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5 

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SECOND  BOOK. 


4 


I 


WALTERS'  ELEMENTARY  GRAPHICS. 


SECOND  BOOK. 


PKOJECTION  DKAWING. 


BY  50HH  DANIEL  Of  A  LIBELS,  ffl.  SG. 

PROFESSOR  OF  INDUSTRIAL,  ART  AND    DESIGNING  IN  THE  KANSAS  STATE 
AGRICULTURAL,    COLLEGE. 


MERCURY  PUBLISHING   HOUSE, 
Manhattan,  Kanwis. 

1894. 


"A  place  should  also  be  found  in  the  school  or  college  course  for  at  least  the  elements  of 
the  modern  synthetic  or  protective  geometry.  It  is  astonishing  that  the  subject  should  be  so 
generally  ignored,  for  mathematics  offers  nothing  more  attractive.  It  posesses  the  con- 
creteness  of  the  ancient  geometry  without  the  tedious  particularity,  and  the  power  of  analyt- 
ical geometry  without  the  reckoning,  and  by  the  beauty  of  its  ideas  and  methods,  illustrates 
the  esthetic  quality  which  is  the  char-m  of  the  higher  mathematics,  but  which  the  elementary 
mathematics  in  general  lacks. "—From  the  report  of  the  committee  on  Secondary  Studies, 
appointed  at  the  meeting  of  the  National  Educational  Association,  July  9, 1892. 


COPYRIGHTED,  1895. 


PREFACE. 


This  course  in  Elementary  Graphics  consists  of  four  separate  text- 
books, each  of  which  is  intended  to  furnish  daily  work  for  a  term  of  from 
six  to  twelve  weeks,  or  its  equivalent.  It  includes  the  subjects  of 

1.  Geometrical  Drawing, 

2.  Projection  Drawing, 

3.  Elements  of  Descriptive  Geometry, 

4.  Linear  Perspective, 

and  was  originally  prepared  for  the  classes  of  the  Kansas  State  Agricul- 
tural (and  Mechanical)  College,  but  has  been  arranged  here  so  that  the 
different  volumes  may  be  used  separately  and  under  varying  conditions. 
It  is  intended,  however,  that  Geometrical  Drawing  should  precede,  and 
Projection  Drawing  should  follow  the  study  of  Plane  and  Solid  Geometry. 

J.  D.  WALTERS. 

Manhattan,  Kansas,  October,  1894. 


M319099 


GLOSSARY 


OF  MATHEMATICAL  AND  TECHNICAL  TERMS. 


The  Numbers  Refer  to  the  Paragraphs  Containing  Definitions  or  Descriptive 

Statements. 


Axial,  non-axial 33 

Axis  of  rotation 13,  14,  15 

Axonometry 3,  31 

Auxiliary  plane 12 

Auxiliary  projection 12 

Black  printing 48 

Blue  printing 48 

Brilliant  point 51 

Brilliant  line 49,  50,  51 

Cabinet  perspective .32 

Conic  section 17,  18,  19,  20,  21 

Co-ordinate 19,  20 

Co-ordinate  planes 12 

Descriptive  geometry 1,  2,  14 

Dip 13 

Directrix 19,  20 

Double  curved  surface 52 

Development 24,  25,  26,  27 

Elevation 7 

Ellipse 17,  18 

Ellipsoid 29 

False  perspective 32 

Focus 17,  18,  19,  20 

Generatrix 18,  19,  20 

Ground  line 9 

Helical 29 

Helical  flange.   29 

Helicoid 29 

Helix   28,  29,  30 

Hyperbola 17,  20 

Inking ...47 


Isometric 3,  31,  35,  36,  37,  38 

Light 39 

Locus 17,  18,  19,  20 

Major  axis 18 

Minor  axis 18 

Monodimetric 31,  32,  33,  34 

Normal 18,  19,  20 

Ordinate 18,  19,  20 

Orthogonal 3,  12 

Orthographic 3 

Parabola 17,  17 

Plan 9 

Planes  of  projection 6,  12 

Plane  of  rotation 13 

Pitch 28 

Projection  6 

Rabattement 15,  22 

Radius  vector 17,  18,  19,  20 

Rotation 13,  17 

Scotia 54 

Screw  line 28,  29,  32 

Secant  plane 15,  10 

Section 15,  16,  17 

Serpentine 37 

Shades 49,  50,  51,  52 

Shadow 39,  40,  41,  42 

Single  curved  surface 49 

Stippling 51 

Torus 29 

Trace 12,  15,  12 

Zincetching 48 


PART  I. 
Definitions,  Problems  and  Exercises. 


INTRODUCTION. 


1.  Projective  Geometry. 

The  objects  of  projective  geometry  are: 

(I.)    To  represent,  by  drawings,  geometrical  magnitudes  in  space. 

(2  )  To  solve  problems  on  forms  in  space  by  construction  in  a  plane, 
and  to  demonstrate  by  the  method  of  projections  the  properties  of  form 
and  position. 

Accordingly  the  subject  matter  of  the  science  may  be  divided  into 

(1.)    Projection  Drawing 

(2.)    Descriptive  Geometry. 

2.  Projection  Drawing. 

Projection  drawing  is  the  art  of  representing  space  forms  upon  plain 
surfaces,  so  as  to  show  their  real  dimensions  and  relations. 

Conceptions  of  the  form  of  solids  or  objects  having  three  dimensions, 
are  at  first  obtained  with  difficulty  from  drawings.  Especially  is  this  the 
case  when  the  drawings  are  not  perspectives  and  lack  shading.  ( Work- 
ing drawings.)  The  systematic  study  of  projection  drawing  is  therefore 
a  matter  of  great  importance  to  everyone  intending  to  follow  an  indus- 
trial, engineering  or  artistic  pursuit.  For  the  same  reason,  the  study  of 
descriptive  geometry  is  usually  preceded  by  a  course  upon  the  methods  of 
rep  resenting  objects  having  three  dimensions.  It  is  evident  that  the 
student  must  learn  to  read  and  draw  the  language  of  space  forms  before 
he  can  expect  to  make  progress  in  the  analysis  or  such  forms,  just  as  he 
had  to  learn  reading  and  writing  before  he  could  commence  the  study  of 
rhetoric  or  literature. 

3.  Different  Methods  of  Projection. 

There  are  at  least  four  scientific  methods  of  projection  drawing: 
(V)    Orthographic  projection. 
(2  )    Axonometric  projection. 
(3  )    Conical  projection. 
(4  )    Spherical  projection. 


WALTERS'  ELEMENTARY  GRAPHICS. 


Each  of  these  has  characteristics  that  make  it  suitable  for  certain 
kinds  of  scientific  or  practical  work.  In  mathematical  and  engineering 
drawing,  the  first,  or  orthographic  method,  is  commonly  used.  This  book 
treats  orthographic  and  axonometric  projection. 

4.    Method  of  Work. 

Experience  has  proved  that  a  series  of  problems  of  progressive  diffi- 
culty, which,  taken  in  their  logical  order,  the  student  can  master  alone, 
or  with  little  assistance  on  the  part  of  the  teacher,  will  accomplish  the 
desired  ends  better  than  any  other. 

Some  of  the  work  in  projection  drawing  offers  better  opportunity  for 
artistic  efforts,  requires  larger  sheets  of  paper,  and  should  be  done  under 
the  eye  of  the  teacher,  while  a  majority  of  the  problems  may  be  solved  on 
smaller  sheets,  simply  with  triangle  and  compasses.  For  this  reason, 
this  book,  like  Book  1,  has  been  divided  into  two  parts: 

Part  I. — Definitions,  problems  and  exercises. 

Part  II.— Shading  and  draughting. 

The  first  part  contains  the  subject  matter  of  the  home  work,  and  the 
second  part  that  for  the  class  room  work.  The  study  of  both  parts  should 
be  commenced  at  the  same  time,  and  should  be  carried  on  together.  One 
day  should  be  given  to  discussions  and  illustrations  of  new  principles  and 
to  blackboard  recitations  by  the  class;  the  next  day  should  be  devoted  to 
draughting  in  the  class  room. 

FIG.  1 


All  home  work  in  projection  drawing  should  be  done  in  ink  and 
should  be  finished  uniformly,  so  as  to  permit  binding  at  the  close  of  the 
term.  Each  plate  should  be  drawn  with  a  heavy  bor«ler  line  inclosing  a 
surface  of  6i  by  9  inches.  Only  one  side  of  the  paper  should  be  used. 
The  sheet  should  not  be  rolled,  but  ought  to  be  carried  "flat"  in  a  large 
book  or  a  portfolio.  "What  is  worth  doing  at  all  is  worth  doing  well !" 
5.  Tools  and  Materials. 

The  tools  required  for  the  home  work  in  projection  drawing  are: 

(1.)  A  pencil  of  rich  quality  and  hardness  that  it  will  take  and  hold 
a  fine  point. 

(2.)    An  eraser. 

(3.)  A  pair  of  solid  white  metal  compasses  with  pencil  and  pen  at- 
tachment. 

(4.)    A  drawing  pen. 

(5.)  A  drawing  board,  12  to  14  inches  bv  20  to  22  inches,  (see  illustra- 
tion.) 


PROJECTION   DRAWING. 


(6.)    A  small  T-square. 

(7.)    A  right  angled  isosceles  triangle  having  sides  of  about  6  inches. 

(8.)    Four  small  white  metal  thumb  tarks. 

Of  materials,  will  be  needed  a  bottle  of  jet  black  writing  ink— not 
writing  fluid,  as  this  would  corrode  the  drawing  pen — and  about  three 
dozen  sheets  of  best  American  drawing  paper,  size  8i  by  11  inches.  This 
paper  is  usually  sold  in  sheets  measuring  17  by  22  inches  and  has  to  be 
folded  and  cut.  If  the  finished  work  is  to  be  bound,  larger  sheets  should 
be  provided  and  the  additional  margin  Should  be  arranged  for  on  the  left 
side  of  each  plate.  Good  grades  of  French  linen  paper  will  also  be  ser- 
viceable. 

The  additional  tools  and  materials  needed  in  shading  and  draughting 
are  described  in  the  Introduction  of  Part  II  of  this  book. 


DIVISION  B. 


GENERAL  PRINCIPLES. 


6.    The  Planes  of  Projection. 

The  solid  or  object  to  be  drawn  is  supposed  to  be  placed  above  a  hori- 
zontal and  in  front  or  a  vertical  plane.  These  planes  are  called  the  planes 
of  projection. 

FIG.  2. 


Perpendiculars,  named  projectors,  are  then  supposed  to  be  dropped 
from  every  corner  or  conspicuous  point  of  the  solid  to  both  planes  of  pro- 
jection. By  connecting  the  ends  of  those  projectors  upon  the  planes  of 
projections  by  lines  that  represent  the  edges  or  outlines  of  the  solid,  two 
pictures  are  obtained. 


WALTERS'  ELEMENTARY  GRAPHICS. 


It  will  be  seen  that  these  pictures,  named  the  horizontal  projection  and 
the  vertical  projection,  form  a  perfect  record  of  the  solid  or  object,  a  record 
that  enables  the  reader  to  imagine  and  measure  the  represented  form. 

In  practical  draughting,  the  horizontal  projection  is  called  plan 
and  the  vertical  projection  elevation. 

A  plan  is  a  drawing  made  on  a  flat  surface,  which  describes  the 
length  and  breadth  of  a  solid  or  object  on  a  surface  that  is  considered  as 
always  lying  horizontal. 

An  elevation  is  a  drawing  made  on  a  flat  surface  that  is  regarded  as 
always  standing  vertical— perpendicular  to  the  plan. 
7.    Plan  and  Elevation. 

In  figure  3,the  student  will  recognize  two  vertical  and  two  horizontal 
projections.!.  e.,the  plans  and  elevations.of  a  dog  house.  A  careful  study 
of  all  the  features  of  this  drawing  will  convince  him  of  the  facts  stated  in 
paragraph  6. 

FIG.  8. 
FRONT  ELEVATION.  SIDE.  ELEVATION. 


FLOOR  PLAN 


ROOF  PLAN. 


SCALE  1  INCH  TO  2  FEET. 


PROJECTION   DRAWING. 


A  carpenter,  asked  so  build  a  dog  house  like  this,  could  go  to  work  at 
once  and  construct  it  without  asking  a  single  question  regarding  its  form. 
Understanding  how  to  use  the  scale  of  one  inch  to  two  feet,  which  has  been 
used  in  making  this  draught,  he  could  measure  the  height  of  the  whole 
object,  the  height  of  the  sides  and  the  height  of  the  opening,  in  the  two 
elevations,  while  he  could  obtain  the  horizontal  measure,  the  thickness 
of  the  boards,  etc.,  in  the  plans. 

To  read  simple  plans  and  elevations,  like  these,  is  not  difficult,  but 
there  are  many  questions  arising  in  connection  with  the  drawing  and 
reading  of  forms  of  three  dimensions  that  will  tax  the  student  more 
severely.  Everyone  can  read  an  item  in  a  book  or  newspaper  without 
much  of  an  effort,  yet  few  are  able  to  write  an  article  grammatically  and 
rhetorically  correct.  In  order  to  acquire  the  ability  to  do  this,  language 
must  be  studied  systematically.  The  student  must  look  for  general 
principles. 

8.  Seven  Principles. 

From  the  given  figures,  and  from  blackboard  drawings  and  models 
exhibited  by  the  teacher,  the  following  principles  may  readily  be  deduced: 

(1  )  Two  planes,  at  right  angles  to  each  other,  are  necessary  to  fully 
represent  the  three  dimensions,  length,  width  and  height,  of  a  solid. 

(2)  These  two  planes  may  really  consist -of  one  surface  only.  By 
dividing  it,  i.  e.,  the  sheet  of  paper,  the  slate,  or  the  blackboard,  by  a 
horizontal  line,  we  can  readily  imagine  the  upper  part  to  be  vertical  and 
the  lower  part  to  be  horizontal. 

This  horizontal  division  line  is  called  the  ground  line. 

(3.)  In  order  that  dimensions  shall  be  seen  in  the  projections  in  their 
true  size  and  relative  position,  they  must  be  parallel  to  that  plane  on 
which  they  are  shown. 

(4.)  Each  plane  shows  two  of  the  dimensions  of  the  solid— the  two 
which  are  parallel  to  it. 

(5.)  The  dimension  which  is  shown  twice,  is  the  one  which  is  parallel 
to  both  of  the  planes. 

(6  )  The  height  of  the  vertical  projection  of  a  point  above  the  ground 
line,  is  equal  to  the  height  of  the  point  itself,  in  space,  above  the  horizon- 
tal plane. 

(7.)  The  perpendicular  distance  of  the  horizontal  projection  of  a 
point  from  the  ground  line,  is  equal  to  the  perpendicular  distance  of  the 
point  itself,  in  front  of  the  vertical  plane. 

9.  Examples  and  Problems. 

Having  established  these  principles,  it  should  not  be  difficult  to  read 
the  following  figures  and  to  construct  the  projections  required  in  this 
paragraph. 

Note  that  the  ground  line  is  marked  G  L  and  is  drawn  in  full  line, 
while  the  projectors,  and  the  edges  not  seen  from  above  and  from  the  front, 
are  drawn  in  dotted  line.  This  should  be  done  in  all  required  work. 


10 


WALTERS    ELEMENTARY   GRAPHICS. 


In  projection  drawing,  to  draw  means  to  tind  the  horizontal  and  verti- 
cal projection.  All  solids  should  be  drawn  with  their  axis  in  vertical 
position  and  standing  upon  H,  unless  otherwise  specified. 

PROBLEM  1.  To  draw  the  projections  of  a  cube  having  edges  of  2f 
inches. 

PROBLEM  2.  To  draw  a  hexagonal  prism  having  a  base  diagonal  of 
2f  inches  and  an  altitude  of  3£  inches. 

PROBLEM  3.  To  draw  a  pentagonal  pyramid,  4  inches  high;  its  base 
having  sides  of  H  inches. 

PROBLEM/!  To  draw  a  triangular  pyramid,  4  inches  high,  with  a 
base  having  sides  of  2  inches.  Represent  it  as  standing  upon  its  apex. 

PROBLEM  5.  To  draw  a  cone  with  its  axis  horizontal  and  parallel  to 
V;  altitude  3  inches;  diameter  of  base  3  inches. 


FIG.  4. 


FIG   5. 


FIG.  6. 


FIG.  7. 


FIG.  8. 


PROBLEM  6.  To  draw  a  sphere  having  a  radius  of  If  inches,  and  hav- 
ing its  center  2  inches  from  both  planes  of  projection. 

PROBLEM?.  To  draw  the  projections  of  a  common  thread  spool, 
consisting  of  a  cylindric  shaft  and  two  truncated  cones,  the  whole  pierced 
by  a  cylindric  opening  Length  of  spool  3  inches,  length  of  shaft  2 
inches,  diameter  of  shaft  1  inch,  diameter  of  ends  2  inches,  diameter  of 
opening  £  inch. 

10.    The  Projection  of  the  Straight  Line. 

From  the  last  exercise  the  student  has  learned  that  a  line  in  space  can 
have  four  radically  different  positions  with  regard  to  the  planes  of  pro- 
jection It  may  be 

(1.)    Parallel  to  both. 

(2.)    Parallel  to  one  and  perpendicular  to  the  other. 

(3.)    Parallel  to  one  and  oblique  to  the  other. 

(4.)    Oblique  to  both. 

If  the  student  will  read  figures  9-14  by  illustrating  them  with  his 
pencil  as  representing  the  line  in  space,  using  the  table  to  represent  H 


PROJECTION   DRAWING- 


11 


and  a  standing  book  to  represent  V,  he  will  notice  that  the  projection  of 
a  line  in  space  is  either 

(a)  equal  to  the  real  line, 

(b)  shorter  than  the  real  line,  or 

(c)  a  point. 

FIG.  9.       FIG.  10.       FIG.  11.       FIG.  12.       FIG.  13.       FIG.  14. 


PROBLEM  8.  To  draw  Mnee  lines  parallel  to  H  and  V,  two  lines  par- 
allel to  V  and  oblique  to  H,  three  lines  oblique  to  H  and  V. 

n.    Mensuration  of  the  Line  in  Space. 

In  the  first  three  cases  of  paragraph  10,  the  line  in  space  may  be 
measured  directly  by  measuring  one  of  its  projections.  In  the  fourth  case 
both  projections  are  shorter  than  the  real  line  and  this  can  therefore  not 
be  measured  directly.  The  length  must  be  tomulindirectly,  which  may  be 
done  in  several  ways. 

From  the  following  figure  it  will  appear  that  the  line  in  space  is  the 
hypotenuse  of  a  triangle  of  which  in  every  case  the  two  sides  are  shown. 
A  FIG.  15.  B 


AB*  =  ta'b').2 
A  B  =  =  VjiTbT 


(m.b) 


(m  b) 


In  construction,  the  real  length  of  the  line  in  space  may  be  found  by 
rotation,  i.  e.,  by  revolving  it  about  a  known  axis  until  it  becomes  parallel 
to  either  of  the  planes  of  projection. 

In  Fig.  B,  the  line  A  B  has  been  revolved  about  the  vertical  axis  A 
M  into  V,  where  it  now  appears  at  full  length.  The  student  will  see,  that 


12 


WALTERS'  ELEMENTARY  GRAPHICS. 


the  line  might  have  been  revolved  about  a  vertical  axis  through  B,  or  a 
horizontal  axis  through  A,  or  a  horizontal  axis  through  B,  or  in  fact, 
about  any  vertical  or  horizontal  axis  in  space.  In  every  case  there  would 
have  been  the  same  result 

The  ability  of  revolving  or  rotating  any  line  in  space  about  any  verti- 
cal or  horizontal  axis  into  a  position  parallel  to  V  or  H  is  very  valuable 
and  the  student  should  execute  such  rotations  until  he  becomes  proficient, 
not  only  in  measuring  lines  of  all  possible  original  positions,  but  also  in 
imagining  such  rotations  in  space,  without  any  drawing  or  experimenting. 


DIVISION  C. 


AUXILIARY  AND  SECANT  PLANES,  ROTATION. 


12.    The  Auxiliary  Plane. 

Figure  2  gives  two  elevations  of  a  dog  house 
resents  the  front  and  the  second  the  side 

Fio  16. 


The  first  elevation  rep- 
Most objects  require  two  eleva- 


tions  on  two  plans,  i.  e.,  they  require  an  additional  projection  upon  an 
auxiliary  plane  of  projection.  Sucn  a  plane  is  usually,  though  not  always, 
a  vertical  plane  placed  perpendicular  to  the  two  main  (co-ordinate)  planes 
of  projection  to  right  or  left  of  the  object. 

An  examination  of  the  above  figure   will  shovv   how  an    auxiliary 
projection  may  be  obtained  from  the  given  main  projections. 


PROJECTION    DRAWING. 


13 


The  dot  and  dash  lines  represent  auxiliary  planes  of  projection,  i.  e., 
they  are  the  intersections  (truces)  of  auxiliary  planes  with  H  or  V.  (These 
two  capital  letters  are  common  aoreviations  for  vertical  plane  of  projec- 
tion and  horizontal  plane  of  projection.) 

After  having  drawn  the  projectors  orthogonal  (rectangular)  to  the 
auxiliary  planes,  these  are  rotated  about  their  vertical  traces  into  V  (or  H, 
as  the  case  may  be)  and  become  a  part  of  V.  The  arcs  show  the  move- 
ment of  the  projections  pending  the  rotation. 

PROBLEM  10.  To  draw  the  horizontal  and  vertical  projection  of  a 
vertical  heptagonal  prism  and  to  find  two  auxiliary  vertical  projections. 

Place  the  right  auxiliary  plane  perpendicular  to  V  and  H,and  the  left 
auxiliary  plane  at  60°  to  V  and  90-  to  £1 .  Draw  the  figures  so  as  to  reason- 
ably fill  the  space  within  the  border  line  of  the  page  (6|  by  9  inches.)  Draw 
all  projectors  in  dotted  line  or  full  red  ink;  the  auxiliary  planes  in  black 
dot  and  dash  line,tbe  G  L  and  the  visible  edges  in  the  projection  sin  strong 
black  line;  the  rear  or  invisible  edges  in  dotted  black  line 
13.  Vertical  Rotation. 

Instead  of  using  the  auxiliary  plane  to  obtain  additional  projections, 
the  object  itself  may  be  revolved  or  rotated  so  as  to  produce  different  pro- 
jections upon  V  or  H. 

FIG.  17. 


Figure  17  shows  the  rotation  of  a  three-sided  pyramid  about  an 
imaginary  axis  The  axis  is  a  line  perpendicular  to  Y  through  point  A. 
The  plane  of  rotation  is  parallel  to  V.  All  points  of  the  pyramid  move  in 
arcs  which  record  as  arcs  in  V,  and  as  straight  lines  parallel  to  G  L  in  H. 
As  a  result  a  second  vertical  projection  is  obtained  which  is  equal  to  the 
first,  but  has  a  different  dip  to  H:  and  a  second  horizontal  projection 
which  is  entirely  different  from  the  first,  but  shows  every  point  at  an  un- 
changed distance  from  V. 


14 


WALTERS'  ELEMENTARY  GRAPHICS. 


Solve  the  following  problem: 

PROBLEM  11.  To  rotate  a  vertical  pentagonal  pyramid  through  an 
angle  of  30°;  the  rotation  to  be  parallel  to  V  and  the  axis  of  rotation  to  be 
i  inch  above  the  apex. 

14.     Horizontal  Rotation. 

In  the  last  paragraph  the  solids  were  rotated  about  horizontal  axes. 
This  paragraph  treats  the  rotation  of  single  solids  about  vertical  axes. 

FIG.  18. 


PROBLEM  12.  To  find  the  projections  of  a  three-sided  pyramid  hav- 
ing a  base  edge  of  2  inches  and  an  altitude  of  3  inches.  The  solid  is  to  be 
drawn  in  four  positions  pending  a  rotation  about  a  vertical  axis,  as 
indicated  in  figure  18. 

Solve  the  following  problem: 

PROBLEM  13.  To  find  the  projections  of  a  cube  having  an  edge  of 
2i  inches,  in  four  positions  pending  a  rotation  about  a  vertical  axis 
placed  in  front  of  the  solid. 

FIG.  19. 


PROBLEM  14.  To  find  the  projections  of  a  right  hexagonal  pyramid 
21  inches  high  and  If  inches  in  diameter  at  its  base,  whose  axis  shall  be 
oblique  to  both  planes  of  projection. 


PROJECTION   DRAWING. 


15 


The  given  illustration  represents  the  method  usually  followed  in" 
placing  an  object  in  an  oblique  position  to  both  planes,  though  descriptive 
geometry  gives  additional  solutions  which  will  be  explained  in  Book  III 
of  this  series.  Figure  A  represents  the  pyramid  standing  on  H  and  par- 
allel to  V.  Figure  B  represents  it  tipped  up  and  back  to  the  left,  and 
moved  forward  parallel  to  V,  s  •  that  every  point  in  it  remains  the  same 
distance  as  before  from  V.  The  vertical  projection  remains  the  same, 
except  that  it  is  oblique  to  G  L.  The  horizontal  projection  is  changed, 
and  can  be  found  by  drawing  projectors  and  parallels.  Figure  C  repre- 
sents the  pyramid,  as  it  is  in  its  second  position,  revolved  45°  about  a 
vertical  axis  of  rotation.  This  places  the  axis  of  the  pyramid  in  the  re- 
quired position  oblique  to  both  planes  of  projection.  The  plan  ia  figure 
C  is  the  same  as  that  of  figure  B,  but  the  vertical  projection  is  changed. 
It  can  be  found  by  drawing  projectors  from  the  third  plan  and  parallels 
to  G  L  from  the  second  elevation. 

FIG.  20. 


Solve  the  following  problem: 

PROBLEM  15.  To  draw  a  plinth  (a  low  square  prism),  first,  as  lying 
upon  H;  second,  as  having  been  pushed  to  the  right  a  distance  equal  to 
its  length,  and  rotated  45°  about  a  horizontal  axis:  third,  as  having  been 
pushed  to  the  right  once  more  and  rotated  30-  about  a  vertical  axis.  Di- 
mensions 1  by  2i  by  2*  inches. 

In  Fig.  17,  an  auxiliary  plane,  U  V  X,  has  been  placed  oblique  to  H 
and  perpendicular  to  V,  so  as  to  coincide  with  one  of  the  slanting  edges 
of  the  pyramid.  As  a  result,  the  student  will  notice  an  additional  pro- 
jection to  the  right  of  the  main  vertical  projection.  How  has  it  been  ob- 
tained? Has  the  oblique  plane  of  projection  been  rotated  about  an  axis? 
Can  you  illustrate  the  movement  by  means  of  models,  i.  e.,  a  solid  and  a 
plane? 


16 


WALTERS    ELEMENTARY    GRAPHICS. 


It  is  frequently  required  to  lay  a  pyramid  or  other  object  having  faces 
oblique  to  the  axis,  so  upon  H  that  it  will  appear  to  rest  upon  one  of  them. 
The  problem  can  be  rapidly  solved  by  this  method. 

15.    The  Secant  Plane. 

A  secant  plane  is  an  imaginary  plane  passing  through  a  solid  or  object 
and  cutting  it  into  two  parts.  The; imaginary  cut  or  surface  is  called 
section.  According  to  the  position  of  the  secant  plane  the  cut  may  give 
a  horizontal  section,  a  vertical  section  or  an  oblique  section. 

The  secant  plane,  like  the  auxiliary,  is  usually  represented  by  its 
traces,  i.  e.,  its  intersections  with  H  and  V.  The  traces  are  drawn  in 
lines  consisting  of  dots  and  long  dashes. 

Figure  21  represents  a  hexagonal  prism  cut  by  a  secant  plane  which 
forms  an  angle  of  45°  with  H,  and  of  90°  with  V.  The  section  is  evi- 
dently an  irregular  hexagon,  the  real  form  of  which  is  found  by  its 
rotation  of  45°  about  a  horizontal  axis  of  rotation  through  A  and  perpen- 
dicular to  G  L.  The  study  of  the  given  figure  will  explain  the  method 
and  reveal  the  real  form. 

FIG.  21. 


The  rotation  of  a  plane  figure  in  space  into  a  plane  of  projection  |s 
called  rabattement. 

Solve  the  following  problem: 

PROBLEM  16.  To  find  the  real  form  of  the  section  of  a  given  triangular 
pyramid,  produced  by  a  secant  plane  forming  an  angle  of  *HJP  to  the  left 
and  passing  through  the  right  base  corner.  ^9 

16.    Sections  of  an  Irregular  Pentagonal  Prism. 

Draw  the  projections  of  an  irregular  pentagonal  prism  resting  with 
one  of  its  faces  upon  H  and  having  its  axis  parallel  to  V.  To  the  left 
place  an  auxiliary  plane  perpendicular  to  G  L.  Upon  this  find  the  aux- 
iliary vertical  projection.  The  figure  is  now  ready  for  the  following 
problem. 


PROJECTION   DRAWING.  17 


PROBLEM  17.  To  find  the  real  form  of  three  characteristic  sections 
of  an  irregular  pentagonal  prism,  produced  by  three  intersecting  planes 
which  intersect  each  other  in  a  line  in  H  and  perpendicular  to  G  L.  One 
section  is  to  be  a  triangle,  one  an  irregular  pentagon,  and  one  a  tra- 
pezium. 

There  are  really  four  characteristic  sections  of  this  prism,  the  fourth 
being  a  rectangle.  The  student  will  readily  see  where  and  how  the 
fourth  section  may  be  obtained. 


DIVISION  D. 


PROJECTIONS  AND  SECTIONS  OR  THE  CYLINDER 
AND  CONE. 


17.    The  Conic  Sections. 

The  cylinder  and  the  cone  are  closely  related  solids.  In  higher 
mathematics  the  first  is  considered  as  being  only  a  special  form  of  the 
second,  i.  e.,  the  cylinder  is  defined  as  being  a  cone  having  an  infinitely 
extended  axis.  For  this  reason  the  cone  will  form  the  chief  object  of 
study  of  this  division. 

FIG.  22. 
ft 


G 


Figure  22  gives  the  vertical  projection  of  a  vertical  cone.  It  also 
shows  the  traces  upon  V  and  through  the  cone  of  a  secant  plane  in  seven 
different  positions.  Every  one  of  these  sections  differs  from  every  other, 
yet  they  are  evidently  related  to  each  other,  because  it  can  easily  be 
shown  how  by  rotation  of  the  intersecting  plane  each  becomes  the  other 
six  by  gradual  changes. 

At  X  assume  an  axis  of  rotation,  perpendicular  to  V.  Then  let  the  in- 
tersecting plane  rotate  on  this  from  position  A  through  B,  C,  D,  E,  F  to  G. 


18  WALTERS'  ELEMENTARY  GRAPHICS. 


The  section  at  A  will  be  a  straight  line. 

The  section  at  B  will  be  an  ellipse. 

The  section  at  C  will  be  a  circle. 

The  section  at  D  will  be  an  ellipse. 

The  section  at  E  will  be  a  parabola. 

The  section  at  F  will  be  an  hyperbola. 

The  section  at  G  will  be  a  straight  line. 

The  sections  at  any  place  between  A  and  C,  and  between  C  and  E, 
will  be  ellipses;  and  the  sections  between  E  and  G,  hyperbolas. 

The  figure  shows  how  the  line,  an  element  of  the  cone  at  A,  is  trans- 
fortned  into  the  other  4  forms— the  circle,  the  ellipse,  the  parabola&nd  the 
hyperbola — and  how  the  last  of  these  forms  again  becomes  a  line  as  the 
plane  is  rotated  out  of  the  cone  at  G. 

By  moving  point  X,  together  with  the  intersecting  planes  A,  B,  C, 
D,  E,  F  and  G,  toward  the  apex  of  the  cone,  another  interesting  trans- 
formation takes  place,  which  culminates,  when  this  point  is  reached, 
into  the  following  series  of  real  or  imaginary  limiting  forms: 

The  circle  becomes  a  point. 

The  ellipse  becomes  a  point. 

The  parabola  becomes  an  element. 

The  hyperbola  becomes  an  isoceles  triangle. 

By  extending  the  axis  of  the  cone,  the  parabola  and  hyperbola  grad- 
ually approach  each  other  and  finally  merge  into  the  same  limiting  form. 
Both  branches  of  both  curves  become  one  cylinder  element. 

A  contraction  (shortening)  of  the  axis  of  the  cone  causes  the  section 
curves  in  space  to  gradually  approach  their  horizontal  projections,  until 
the  limiting  form  of  the  hyperbola,  the  straight  line  (chord),  is  reached. 

The  student  should  convince  himself  of  these  interesting  relations  of 
the  different  sections  of  the  cone  by  carefully  experimenting  with  a  large 
cone,  upon  the  curved  surface  of  which  he  should  draw  crayon  lines  rep- 
resenting the  sections  named  in  this  paragraph.  He  should  make  an 
effort  to  imagine  the  horizontal  and  vertical  projections  of  these  sections 
by  looking  at  the  model  from  different  positions,  and  should  familiarize 
himself  with  the  names. 

The  conic  section  curves  appear  also  as  loci  apart  from  the  cone,  but 
the  cone  is  the  most  convenient  form  to  observe  their  relation  to  each 
other. 

In  order  to  study  the  mathematical  character  of  these  figures  or 
sections,  it  will  be  well  to  lake  them  up  separately,  leaving  out  the  circle, 
which  has  been  studied  to  some  extent  in  plain  and  solid  geometry. 

18.    The  Ellipse. 

If  a  point  moves  in  such  a  manner  that  the  sum  of  its  distances  from 
two  fixed  points  (foci)  is  constant,  the  locus  traced  by  the  moving  point  is 
an  ellipse. 


PROJECTION  DRAWING. 


19 


Figure  23  illustrates  this  definition  and  shows  how,  by  means  of  two 
pins,  a  string  and  a  pencil,  an  ellipse  may  he  drawn. 

The  ellipse  has  two  foci  at  A  and  B.  It  is  a  symmetrical  curve  about 
two  lines,  one  of  which  is  called  major  axis  and  contains  the  foci,  and 
another,  which  is  called  the  conjugate  or  minor  axis.  The  axes  are  mutu- 
ally perpendicular  and  bisect  each  other.  The  extremities  of  the  major 
axis  are  called  vertices.  Any  line  from  a  focus  to  the  ellipse  is  called 
radius  vector.  Any  line  perpendicular  to  the  major  axis  is  called  ordinate. 
A  tangent  to  an  ellipse  touches  but  does  not  cut  the  curve.  A  normal  is  a 
perpendicular  to  a  tangent  at  the  point  of  contact;  it  bisects  the  angle 
formed  by  the  radii.  The  sum  of  the  radii  vector  of  the  moving  point 
in  every  position  is  equal  to  the  major  axis. 

FIG.  23.  FIG.  24. 


x^ 


PROBLEM  17.    (Figs.  23  and  24.)    To  construct  an  ellipse. 

(1 )  its  major  axis  and  one  focus,  or 

(2)  its  major  and  minor  axis,  or 

(3)  its  minor  axis  and  one  focus,  or 

(4)  its  two  axes,  or 

(5)  its  two  foci  and  one  point  in  the  ellipse  being  given. 

The  given  definition  and  statements  of  mathematical  qualities  of  the 
ellipse  show  that  the  problem  is  really  the  same  in  all  five  cases.  In  all 
cases  it  is  necessary  to  find  first  the  two  foci  and  the  length  of  the  major 
axis,  i.  e.,  the  sum  of  the/ocaZ  distances. 

After  having  found  the  foci  and  the  major  axis,  the  problem  may  be 
solved  either  with  the  string  or  the  compasses. 

In  figure  23,  pins  are  inserted  at  the  foci  and  a  string  or  a  thread  is 
tied  there,  equal  in  length,  between  the  pins,  to  the  major  axis,  or,  which 
is  the  same,  reaching  from  one  focus  to  the  extremity  of  the  minor  axis 
and  back  to  the  other  focus.  The  inserted  pencil  will  then  describe  the 
required  locus. 

In  figure  24,  the  major  axis  is  cut  at  random  as  shown.  With  A  1, 
A  2,  A3,  as  the  radii  and  F  and  F'  as  centers,  arcs  are  described  in  each 
quadrant.  With  B  1,  B  2,  B  3,  etc.,  as  tlie  radii,  and  F  and  F'  as  centers, 
these  arcs  are  intersected.  A  free-hand  curve  drawn  through  the  points 
of  intersection  will  be  the  required  ellipse. 


20 


WALTERS'  ELEMENTARY  GRAPHICS. 


There  are  many  other  methods  of  constructing  the  ellipse.  Some  of 
these  are  simple,  (compare  figures  30,  32, 48, 49  and  59)  and  others  are  com- 
plex, but  all  involve  the  knowledge  of  some  branch  of  analytic,  or  pro- 
jective  geometry,  which  the  student  may  not  have  studied. 

19    The  Parabola. 

The  parabola  is  a  curve  which  at  all  points,  is  equally  distant  from  a 
fixed  point  (focus,)  and  from  a  fixed  straight  line  (directnx.)  Figure  25 
represents  a  parabola  with  the  directrix  (M  N,)  the  focus  (F,)  the  axis,  an 
ordinate,  three  radii,  three  co-ordinates,  a  tangent,  a  normal,  and  a  chord. 

Find  these  lines  and  note  that  the  radius  vector  of  every  point  in  the 
curve  is  equal  to  the  co-ordinate  of  the  same  point. 

The  parabola  is  sometimes  defined  as  an  infinitely  long  ellipse.  Fig. 
22  shows  how  by  gradual  change  of  the  angle,  which  the  intersect- 
ing plane  forms  with  the  axis  of  the  cone,  the  elliptic  section  becomes 
longer  until  the  parabola  is  approached.  The  ellipse  intersects  all  ele- 
ments of  the  cones  of  which  it  is  a  section.  The  parabola  intersects  all 
elements  except  one — the  one  which  is  parallel  to  it. 

The  circle  may  be  considered  as  one  limiting  form  of  the  ellipse,  the 
parabola  as  a  second,  and  the  straight  line  as  a  third. 

FIG.  26. 


PROBLEM  18.  (Fig.  25.)  Given,  the  directrix  and  focus,  to  describe 
the  curve  of  a  parabola. 

Assume  a  directrix  M  N  and  a  focus  F.  Draw  the  axis  perpendicular 
to  the  directrix.  Bisect  F  A.  This  is  evidently  a  point  in  the  curve, 
being  equidistant  from  directrix  and  focus.  Next  draw  a  series  of  in- 
definite ordinates,  on  both  sides  of  the  axis,  and  find  points  upon  them 
in  the  following  manner:  Measure  1  A  and  with  the  focus  as  center, 
cut  the  ordinate  in  H.  Then  the  distance  of  F  H  is  equal  to  the  distance 
from  II  to  the  directrix.  It  is  therefore  a  point  in  the  required  curve. 
In  a  similar  manner,  a  series  of  additional  points  may  be  found,  which, 
connected  by  a  free-hand  curve,  will  give  the  required  parabola. 


PROJECTION   DRAWING. 


21 


PROBLEM  19.  (Fig.  26  )  To  inscribe  a  parabola  within  a  given  rect- 
angle. 

The  figure  shows  that  the  parabola  consists  of  diagonals  in  the  trape- 
zoids  formed  within  the  given  rectangle  by  the  two  sets  of  connecting 
lines. 

20.    The  Hyperbola, 

If  a  point  moves  in  such  a  manner  that  its  distance  from  a  fixed  point 
is  always  greater  than  its  perpendicular  distance  from  a  fixed  straight 
line,  in  a  constant  ratio,  the  curve  traced  by  the  moving  point  is  an 
hyperbola. 

The  fixed  point  is  the  focus,  and  the  fixed  straight  line  is  the  directrix. 

In  other  words,  an  hyperbola  is  a  curve  that  is  equidistant  at  every 
point  from  two  unequal  circles.  See  figure  28 

The  properties  and  functions  of  the  hyperbola  are  similar  to  those  of 
the  ellipse  and  the  parabala. 


FIG.  27. 


FIG-.  28. 


PROBLEM.  20.    To  inscribe  an  hyperbola  within  a  given  rectangle. 
The  solution  will  be  plain  from  a  study  of  figure  27. 
PROBLEM  21.    To  draw  a  plane  curve  equidistant  from  two  given 
circles. 

The  solution  will  be  apparent  from  figure  28. 

21.    Projection  of  the  Circle,  the  Cylinder  and  the  Cone. 

A  circle  may  assume  four  radically  different  positions  to  the  planes 
of  projection.    It  may  be 

(1.)    Perpendicular  to  both. 

(2.)    Perpendicular  to  one  and  parallel  to  the  other. 

(3. )    Perpendicular  to  one  and  oblique  to  the  other. 

(4.)    Oblique  to  both. 

In  case  (1)  both  projections  are  straight  lines. 


22 


WALTERS    ELEMENTARY   GRAPHICS. 


In  case  (2)  one  projection  is  a  line  and  the  other  a  circle. 

In  case  (3)  one  projection  is  a  line  and  tha  other  an  ellipse. 

In  case  (4)  both  projections  are  ellipses. 

In  all  cases  the  greatest  length  of  the  projection  is  equal  to  the  di- 
ameter of  the  represented  circle,  because  in  every  position  of  the  circle 
one  of  its  diameters  must  be  parallel  to  each  plane  of  projection. 

If  the  projection  is  an  ellipse,  the  length  of  the  minor  axis  depends 
on  the  angle  which  the  plane  of  the  circle  in  space  forms  with  the  plane 
of  projection.  In  case  (4)  the  minor  axes  of  the  two  ellipses  may  be,  but 
are  not  necessarily,  equal. 

FIG.  29. 


Having  thoroughly  comprehended  these  elementary  principles,  and 
illustrated  every  case  by  means  of  a  circular  disc  and  two  coordinate 
planes,  the  projection  of  the  cylinder  and  cone  should  not  be  difficult. 
Figure  29  represents  a  horizontal  cylinder  lying  upon  H,  with  its  axis 
oblique  to  V.  This  horizontal  projection  is  a  rectangle,  and  must  be 
drawn  first.  The  vertical  projection  is  obtained  from  the  horizontal  pro- 
jection as  follows:  Project  the  centers  of  the  circles  into  V  and  draw 
the  top  element.  The  end  circles  of  the  cylinder  will  appear  as  two  equal 
ellipses  in  V.  Find  their  axes  by  drawing  the  necessary  projectors. 
Having  found  these,  the  ellipses  may  be  drawn  as  shown  in  figure  23  or 
24. 

Another  method  consists  in  rotating  the  rear  circle  into  V,  about  a 


PROJECTION   DRAWING. 


23 


vertical  axis  Z,  marking  a  number  of  points  in  it  as  C,  D,  E  and  F.and  re- 
volving it  back  again  to  C  Z.  Pending  this  revolution,  or  counter  rota- 
tion, as  it  is  called,  each  point,  except  Z,  describes  a  horizontal  arc  in 
space,  which  can  easily  be  drawn  in  II  and  V  Parallels  to  G  L  and  pro- 
jectors will  give  the  positions  of  the  marked  points  in  the  required 
vertical  projections.  The  outline  is  drawn  freehand  through  the  estab- 
lished points.  The  ellipse  representing  the  front  circle  can  be  be  drawn 
in  the  same  manner. 

The  student  will  recognize  in  this  work  a  method  of  constructing  an 
ellipse  of  given  dimensions. 

Solve  the  following  problems: 

PROBLEM  22.  To  find  the  projections  of  a  cylinder  oblique  to  H  and 
parallel  to  V. 

PROBLEM  23.  To  find  the  projections  of  a  cylinder  of  given  dimen- 
sions, oblique  to  both  planes  of  projections. 

The  problem  is  to  be  solved  like  problem  15,  figure  20. 

22.    The  Elliptic  Section  of  the  Cylinder  and  Cone. 

The  study  of  figures  30  and  31,  which  illustrate  two  simple  methods 
of  obtaining  the  real  form  of  elliptic  sections  of  the  cylinder,  ought  to 
be  sufficient  to  enable  the  student  to  find  the  real  form  of  elliptic  sec- 
tions of  the  cone  also 


FIG   30 


FIG.  31 


The  first  method  rotates  the  intersecting  plane  into  H  and  obtains 
the  required  ellipse  as  as  a  horizontal  projection.  The  rotation  is  like 
that  of  figures  21  and  29  and  is  called  rabattement. 

The  second  method  rotates  the  ellipse  on  its  major  axis  into  V-  The 
ordinates,  i.  e.,  the  perpendiculars  to  the  major  axis  are  obtained  by 
measuring  the  chords  in  the  liorizonial  section  which  is  a  circle. 

Solve  the  following  problems: 

PROBLEM  24.    Find  the  horizontal  projection  and  the  real  form  of 


24 


WALTERS7   ELEMENTARY    GRAPHICS. 


an  elliptic  section  of  a  given  cone.    The  real  form  is  to  be  found  by  the 
first  method. 

PROBLEM  25.  Find  the  horizontal  projection  and  the  real  form  of 
an  elliptic  section  of  a  given  cone.  The  real  form  is  to  be  found  by  the 
second  method. 

23.    Parabolic  and  Hyperbolic  Sections. 

Both  sections  are  frequently  met  with  in  mechanical  work;  it  is 
impossible,  however,  to  learn  to  draw  them  correctly  and  rapidly  in  all 
different  positions  without  the  knowledge  of  analytic  geometry,  which 
teaches  their  mathematical  properties  and  relations  as  plain  geometry 
teaches  the  properties  and  relations  of  the  circle.  No  student  who  would 
master  drawing  can  afford  to  neglect  the  study  of  this  highly  important 
branch  of  mathematics. 

FIG.  32. 


Figure  32  shows  how  to  find  an  hyperbola  from  its  given  projections. 
The  cone  is  posed  so  that  both  projections  of  the  section  are  straight 
lines.  The  drawing  of  a  number  of  elements  will  locate  points  that  can 
be  determined  in  all  required  positions. 

Solve  the  following  problems: 

PROBLEM  26.  Given  the  projections  of  a  cone  and  the  horizontal 
projection  of  an  hyperbolic  section,  to  find  the  vertical  projection  and 
the  real  form  of  the  section. 

PROBLEM  27.    Given  the  vertical  projection  of  a  cone  and  the  verti- 


PROJECTION    DRAWING. 


25 


cal  projection  of  a  parabolic  section,  to  find  the  horizontal  projection  of 
the  cone  and  the  parabola,  and  to  find  the  real  form  of  the  parabola. 

PROBLEM  28  Given  tha  vertical  projection  of  a  cone  with  the  verti- 
cal projection  of  a  parabolic  section,  to  find  the  vertical  projection  after 
a  rotation  of  4-5°  about  the  axis  of  the  cone. 


DIVISION   E. 


DEVELOPMENT  OR  SURFACES. 


34    The  Development  of  Rectilinear  Surfaces. 

If  a  surface  of  a  geometric  solid  be  rolled  upon  a  plane  so  as  to  bring 
each  consecutive  face,  or  each  consecutive  element  if  the  solid  is  a  single 
curved  surface,  in  contact  with  that  plane  without  stretching,  folding  or 
tearing  the  face  or  element,  the  surface  is  said  to  be  developed,  and  the 
plane  figure  which  results  therefrom  is  termed  its  development. 

The  cube,  the  rectangular  block,  the  prism,  the  pyramid,  and  all 
polyhedrons  are  developable. 

Figures  33  and  34  represent  the  developed  surface  of  a  right  hex- 
agonal prism  and  a  right  hexagonal  pyramid,  and  explain  how  the 
development  has  been  obtained. 


FIG.  33. 


FIG.  34. 


Solve  the  following  problems:    ' 

PROBLEM  29.  To  find  the  development  of  a  cube  having  a  right 
square  pyramid  upon  two  opposite  faces. 

PROBLEM  30.  To  find  the  development  of  a  truncated  pentagonal 
pyramid. 

PROBLEM  31.  To  find  the  development  of  a  cube  cut  into  halves  by 
any  oblique  plane,  and  having  sides  of  1  inch. 

PROBLEM  32.    To  find  the  development  of  a  tetrahedron. 


26  WALTERS'  ELEMENTARY  GRAPHICS. 


PROBLEM  33.    To  find  the  development  of  an  octahedron. 
25.    The  Development  of  the  Cylinder. 

The  development  of  a  right  cylinder  is  an  oblong  or  square,  having  a 
height  equal  to  the  axis  or  an  element,  and  a  base  equal  to  the  circum- 
ference. In  practical  work  of  ordinary  character  the  circumference 
is  usually  measured  by  dividing  the  base  circle  into  twelve  equal  parts 
and  measuring  the  chord  of  such  an  arc  of  30°  twelve  times.  The  result- 
ing perimeter  is,  of  course,  slightly  less  than  the  circumference,  because 
the  chord  is  shorter  than  its  arc. 

In  plane  geometry  the  student  has  learned  that  the  ratio  of  the  di- 
ameter of  the  circumscribed  circle  to  the  perimeter  of  the  inscribed 
dodecagon  is  1  :  3.1058. .,  while  the  ratio  of  the  diameter  to  the  circle  is 
1 : 3.1416. .  The  above  method  of  approximation  is,  therefore,  nearly  cor- 
rect. 

Solve  the  following  problem: 

PROBLEM  34.  To  find  the  development  of  a  cylinder  having  an  axis 
of  3  inches  and  a  diameter  of  2f  inches. 

PROBLEM  35.  To  find  the  development  of  a  rectangular  stove-pipe 
elbow  consisting  of  two  equal  pices  of  sheet  iron  Length  of  axis  of  each 
part,  24  inches;  diameter  of  pipe,  8  inches;  scale,  1  inch  to  4  inches. 

NOTE.— The  tinsmith  cuts  both  parts  out  of  one  piece  of  sheet  iron 
measuring  about  26  by  48  inches.  How  can  he  do  this  V 

26    The  Development  of  the  Cone. 

The  development  of  the  curved  surface  of  a  co1  e  is  a  sector. 

Solve  the  following  problems  for  which  no  figures  or  solutions  are 
given: 

PROBLEM  36.  To  draw  the  patterns  required  by  the  tinner  in  mak- 
ing a  funnel  consisting  of  a  combination  of  two  truncated  cones.  Axis 
of  large  cone,  4  inches;  axis  of  small  cone,  3  inches;  diameters  of  large 
cone,  5  inches  and  1  inch;  diameters  of  small  cone,  1  inch  and  finch. 
No  handle. 

PROBLEM  37.  To  find  the  development  of  a  cone  truncated  by  an 
oblique  plane.  Axis  of  complete  cone,  5  inches;  axis  of  truncated  part, 
2J  inches;  diameter  of  base,  3i  inches;  slant  of  section,  30°  to  H. 

27.    Miscellaneous  Problems  Pertaining  to  Development. 

A  double  curved  surface  is  one  that  has  no  straight  elements,  like  the 
surface  of  the  sphere,  the  torus,  the  ellipsoid,  etc.  Such  a  surface  can  not 
be  fully  developed.  All  that  can  be  done  is  to  find  a  close  approxima- 
tion by  dividing  the  surface  by  means  of  meridians  into  a  number  of 
lunes  and  arranging  these  on  a  plane.  The  larger  the  number  of  lunes  is 
made,  the  less  stretching  or  folding  there  will  be.  Tinners,  who  are 
often  required  to  construct  such  forms  out  of  tin,  sheet  iron,  sheet  zinc, 
or  sheet  copper,  seldom  use  more  than  12  lunes. 

PROBLEM  38.    To  draw  the  covering  for  a  sphere. 


PROJECTION  DRAWING. 


27 


Draw  the  elevation  or  plan  of  the  solid.  Divide  the  circumference 
into  12  equal  parts  and  lay  these  off  on  a  straight  line.  With  a  radius  of 
9  of  these  parts  draw  a  right  and  left  arc  through  each  division  point, 
except  the  end  points  which  require  only  one  arc.  These  arcs  will  inter- 
sect above  and  below  the  straight  line  and  form  the  required  twelve  lunes 
—  an  approximate  covering  for  the  sphere. 

PROBLEM  39.  To  draw  the  covering  for  a  sphere  having  a  diameter 
of  4  inches.  The  covering  to  consist  of  four  zones  of  equal  width. 

Tinners  and  coppersmiths  sometimes  construct  globes  of  zones  in- 
stead of  lunes.  When  copper  or  some  other  highly  malleable  metal  can 
be  used,  a  little  hammering  of  the  four  conic  surfaces,  which  form  the 
zones  will  give  the  required  sphere. 


DIVISION  F. 


THE  HELIX. 


28.    General  Principles  of  the  Helix, 

The  curves  may  properly  be  divided  into  plane  curves,  such  as  the 
circle,  the  conic  sections,  the  spiral,  the  trochoids,  etc.,  and  into  space 
curves,  such  as  the  he lix  or  screw  line  which  forms  the  subject  of  investi- 
gation of  paragraphs  28,  29  and  30. 

A  helix  is  the  locus  of  a  point  which  moves  along  the  surface  of  a 
cylinder  in  such  a  way  that  a  constant  ratio  is  maintained  between  the 
measure  of  its  rotation  and  ascent. 

FIG.  40. 


It  is  sometimes  defined  as  a  spiral  line  that  constantly  advances  in  the 
direction  of  a  straight  line  called  its  axis.  The  student  who  has  mastered 
the  principles  of  Division  E  may  assist  his  mind  in  imagining  this  line 


28 


WALTERS     ELMENTARY   GRAPHICS. 


by  developing  the  curved  surface  of  a  cylinder,  drawing  a  diagonal  or 
any  oblique  line  in  the  development,  and  winding  this  again  upon  the 
solid.  After  the  rewinding  the  diagonal  becomes  a  helix. 

The  helix  is  sometimes  called  screwline,  because  it  is  found  in  the 
thread  of  every  screw.  See  figures  35,  36,  37,  38  and  39. 

If  the  curve  ascends  and  passes  from  left  to  right  in  front  of  the  axis, 
when  the  axis  is  vertical,  the  helix  is  said  to  be  right-hand,  and  if  it 
passes  from  right  to  left  it  is  said  to  be  left  hand. 

The  pitch  of  a  helix  is  the  distance  it  advances  along  the  axis  in  mak- 
ing one  revolution. 

FIGS.  35,  36,  37,  38,  39. 


Solve  the  following  problems: 

PROBLEM  40.  (Fie.  37).  To  draw  the  projections  of  a  right-hand 
/•  helical  line  2  inches  in  diameter,  5  inches  long,  and  3  inches  pitch. 

PROBLEM  41.  To  draw  a  left-hand  helix  having  the  same  dimen- 
sions and  pitch  as  that  of  the  last  problem. 

Careful  study  of  figure  40.  which  is  drawn  I  the  required  scale,  will 
reveal  the  method  of  construction.  The  helix  is  draw  n  free-hand,  through 
the  located  points. 


PROJECTION   DRAWING. 


29 


(2.) 
(3.) 
(4.) 
(5.) 


29.    The  Helical  Band,  the  Helical  Flange,  the  Helicoid. 

u       ^     •*    *    -     * 

All  surfaces  may  be  divided  into: 
(1.)    Planes, 

Single  curved  surfaces  (cylinder,  cones), 
Double  curved  surfaces  (sphere,  torus,  ellipsoid), 
Warped  surfaces  (helical  flange,  helicoid), 
Irregular  surfaces. 
One  of  the  surfaces  named  in  the  headline  of  the  this  paragraph  be- 
longs to  the  second,  and  two  belong  to  the  fourth  class. 

A  helical  band  is  the  surface  ota  cylinder  between  equal  and  parallel 
helices.  See  figures  35,  38  and  42 

A  helical  flange  is  a  warped  surface  that  is  between  two  parallel  but 
unequal  helices  and  is  perpendicular  to  their  common  axis.  See  figure 
41. 

FIG.  41. 


A  helicoid  is  a  warped  surface  that  is  between  two  parallel  but  un- 
equal helices  and  is  at  an  invariable  acute  angle  t  >  their  axis.  See  figure 
42. 

It  may  be  said  that,  as  the  circle  and  the  straight  line  are  the  limit- 
ing forms  of  the  ellipse,  the  helical  flange  is  a  limiting  form  of  the 
helicoid. 

A  serpentine  is  a  bent  cylinder  the  axis  of  which  is  a  helix.    It  is  some 
times  defined  as  the  locus  of  a  sphere,  the  center  of  which  moves  along  a 
helix.    The  bed-spring  and  the  cork  screw  furnish  illustrations  of  this 
highly  interesting  space  form.    The  groove  of  figure  37  is  a  semi  set-pen 
tine.    So  is  the  groove  of  the  drill  described  in  problem  47 


30 


WALTERS'  ELEMENTARY  GRAPHICS. 


Solve  the  following  problems: 

PROBLEM  42.  To  find  the  projections  of  a  right-hand  helical  band, 
i  inch  wide.  Diameter  of  cylinder,  3  inches;  pitch,  H  inch* 

PROBLEM  43  To  draw  the  projections  of  a  left-hand  helical  flange 
4  inch  wide,  2  inches  in  outer  diameter,  4  inches  long,  and  H  inch  pitch. 

PROBLEM  44.  To  draw  the  projections  of  a  left-hand  helicoid  screw. 
Diameter  of  outer  helix,  3  inches;  diameter  of  inner  helix,  2  inches; 
pitch,  1  inch.  Compare  with  figure  42. 

PROBLEM  45.  To  draw  a  right-hand  flange  screw  of  two  threads 
(jack  screw).  Diameter  of  the  four  outer  helices,  4  inches;  diameter  of 
the  four  inner  helices,  3  inches;  pitch  of  each  thread,  2  inches.  Compare 
figure  41. 

FIG.  42. 


30.    Additional  Problems  on  the  Helix. 

PROBLEM  46.  To  find  the  approximate  projections  of  a  three-inch 
rope  consisting  of  3  equal  strands.  Pitch,  6  inches. 

PROBLEM  47.  To  find  the  approximate  projections  of  a  drill  with  a 
conical  point. 

A  drill  is  simply  a  steel  cylinder  with  a  dull  conical  point,  and  a 
slightly  enlarged  square  rear  end  for  fastening  it  in  the  brace.  It  has 
one  helical  groove  of  two  revolutions  and  about  21  inches  pitch.  The 
groove  extends  from  the  square  rear  end  to  the  conical  point  and  is  semi- 
cylindric  at  every  point.  Compare  figure  37. 


PROBLEM  48.    To.find  the  projections  of  a  helical  stairway  winding 


PROJECTION  DRAWING.  31 


about  a  column  of  2  feet  in  diameter.  Distance  from  floor  to  floor  10 
feet,  8  inches;  height  of  riser,  8  inches;  diameter  of  whole  stairwell,  18 
feet.  Handrails,  nosing  and  nosing  moulds  not  required. 

PROBLEM  49.    To  draw  a  helix  upon  a  cone. 

With  the  exception  of  this  problem  the  subject  of  conic  helices  has 
been  omitted  here  as  being  beyond  the  reach  of  elementary  graphics. 


DIVISION  G. 


AXONOMETRIC  PROJECTION. 


31.  General  Principles  of  Axonometric  Projection. 

The  three  imaginary  straight  lines  which  represent  the  three  dimen- 
sions, length,  breadth  and  height  of  an  object  are  called  the  axes  of 
dimension.  All  trained  geometricians  imagine  these  axes  as  intersecting 
each  other  at  right  angles  at  the  center  of  the  solid,  object  or  bulk  to  be 
measured. 

In  axonometric  projection  the  object  is  drawn  so  that  two  or  all  axes 
of  dimension  form  equal  angles  with  the  vertical  plane  of  projection, 
while  the  horizontal  plane  of  projection  is  dispensed  with  entirely. 

When  a  solid  or  object  is  posed  so  that  two  of  its  axes  of  dimension 
are  parallell  to  V  and  the  third  perpendicular  to  V,  it  is  said  to  be  in  a 
monodimetric  position,  and  when  it  is  placed  so  that  all  three  axes  form 
equal  angles  with  V,  it  is  said  to  be  in  an  isometric  position.  There  are, 
therefore,  two  systems  or  methods  of  axonometric  projection: 

(1.)    Monodimetric  projection. 

(2.)    Isometric  projection. 

32.  Monodimetric  Projection. 

In  monodimetric  projection  the  object  is  placed  so  that  its  principle 
arid  most  characteristic  face  is  parallel  to  the  plane  of  projection,  and 
can  be  drawn  as  it  is.  All  other  lines  that  are  parallel  to  the  principal 
face  lines  are  also  projected  at  their  real  lengths  and  in  their  real  posi- 
tions, while  the  lines  that  form  right  angles  with  the  face,  i.  e.  the  lines 
parallel  with  the  third  axis  of  dimension,  are  drawn  at  angles  of  4-1  to 
the  horizontal  and  vertical  lines  of  the  front  face  either  to  right  or  left, 
up  or  down.  The  length  of  such  a  retreating  line  is  made  equal  to  the  shortest 
side  of  a  triangle  having  angles  of  105^,  4-5^.  3(P,  and  whose  longest  side  equals 
he  line  to  be  represented.  A  few  worked  problems  will  explain  this  some- 
what complex  statement. 


32 


WALTERS'  ELEMENTARY  GRAPHICS. 


PROBLEM  50.  (Fig.  43).  To  draw  the  monodimetfic  projection  of  a 
cube  of  given  dimension. 

Draw  A  B  equal  to  the  given  edge.  Draw  the  square  ABED. 
Draw  C  A  at  3(P  to  A  B,  and  B  C  at  45°  to  A  B.  The  angle  at  0  will  evi- 
dently measure  105°  Draw  the  remaining  edges,  representing  those  not 
visible  by  dotted  lines. 

FIG.  43.  FIG.  44. 

B  % 


PROBLEM  51.  (Fig.  44).  To  draw  the  monodimetric  of  a  hori- 
zontal cylinder. 

Locate  A  and  draw,  full  size,  the  face  circle.  Draw  A  B  equal  to  the 
length  of  the  required  axis.  Draw  C  B  at  30°  and  A  C  at  45°  to  A  B. 
With  C  as  the  center  draw  the  rear  circle.  Add  the  outline  elements. 

Solve  the  following  problems. 

PROBLEM  52.  To  construct  the  monodimetric  of  a  horizontal  hex- 
agonal prism  having  a  base  side  of  li  inch  and  an  axis  of  31  inches. 

It  is  evident  that  this  solid  could  be  drawn  in  four  different  posi- 
tions. Its  lateral  edges  could  be  drawn  so  as  to  retreat  to  right  and 
upward,  to  right  and  downward,  to  left  and  upward,  to  the  left  and  down- 
ward. Select  any  of  the  three  latter  positions. 

PROBLEM  53.  To  draw  the  monodimetric  of  a  grindstone ;  radius, 
2i  inches ;  thickness,  i  inch ;  diameter  of  square  hole  at  the  center,  i 
inch. 

No  conditions  as  to  position  being  given,  it  is  evident  that  the  grind- 
stone may  be  drawn  in  any  monometric  position.  What  are  these 
positions  ?  Draw  the  object  in  one  of  the  four  positions  stated  above. 

PROBLEM  54.  To  draw  the  monodimetric  of  a  waterbucket  having 
an  axis  of  1  foot,  abase  diameter  of  10  inches,  atop  diameter  of  14  inches, 
and  a  thickness  in  every  part  of  1  inch.  Add  two  1-inch  hoops  and  a 
semicircular  wire  handle.  Finish  the  drawing  neatly. 

33.    Monodimetric  Projection  of  Non-Axial  Lines. 

The  following  worked  problem  is  intended  to  answer  the  question  of 
how  to  represent  and  measure  lines  that  are  not  parallel  to  axes  of  di- 
mension. 


PROJECTION    DRAWING. 


33 


PROBLEM  55.  (Fig.  45).  To  find  the  monodimetric  of  a  horizontal 
pentagon. 

Draw  a  diagram  of  the  figure  (Fig.  32  or  36,  Vol.  I)  and  in  it  draw 
the  lines  5  6  and  3  4.  Draw  A  B  equal  to  1  2.  Draw  7  D  at  45°  to  A  B, 
and  make  it  equal  to  5  6  by  ^|33.  Find  point  F  in  the  same  names. 
Make  F  E  and  F  C  equal  to  the  halves  of  34.  Draw  the  outline. 

By  drawing  five  equal  vertical  lines  at  the  angular  points  of  the 
monodimetric  pentagon,  and  by  connecting  the  upper  ends  of  these  verti- 
cals, the  monodimetric  of  a  pentagonal  prism  is  obtained. 

Still  another  monodimetric  projection  may  be  obtained  by  drawing 
the  lateral  edges  of  the  prism  downward,  making  the  number  of  possible 
cases  not  four,  but  six. 

Fio.  45. 


Solve  the  following  problems: 

PROBLEM  56.  To  find  the  monodimetric  of  a  vertical  heptagonal 
pyramid. 

PROBLEM  57.  To  find  the  monodimetric  of  a  truncated  triangular 
pyramid.  Height  of  frustrum.  3  inches;,  side  of  base,  3  inches;  side  of 
section,  2  inches 

It  is  difficult  to  find  the  monodimetric  of  a  horizontal  circle.  When 
a  circular  object  is  to  be  drawn,  it  should  be  posed  so  that  the  circles  will 
be  parallel  to  V. 

34.    Additional  Problems  in  Monodimetric  Projection. 

Monodimetric  projection  is  much  used  in  drawing  details  of  stone 
and  lumber  construction,  such  as  mortise-joints,  dovetail  joints,  miter- 
joints,  stretcher- joints,  building  hardware,  brick  and  stone  arches,  etc  It 
is  therfore  often  called  shop  perspective,  false  perspective,  parallel  perspect- 
ive, or  cabinet  perspective.  In  as  much  as  a  monodimetric  projection  shows 
the  front  and  two  sides  of  the  object,  it  resembles  a  true  perspective  pict- 
ure; but  it  is  not  drawn  in  accordance  with  the  laws  of  linear  perspective, 
and  these  terms  are  entirely  improper  and  confusing  The  student 
should  not  use  them. 


34 


WALTERS'  ELMENTARY  GRAPHICS. 


Solve  the  following  problems: 

PROBLEM  58.  To  draw  the  monodimetric  of  a  table,  foot-stool,  or 
tool  bench. 

PROBLEM  59.  To  draw  the  monodimetric  of  any  three  of  the  Egyp- 
tian letters  of  fl  47,  enlarging  them  to  four  diameters  and  giving  them  a 
thickness  of  i  inch. 

PROBLEM  60.  To  draw  the  monodimetric  of  a  dovetailed  box  with  a 
half-open  sliding  lid.  Add  the  scale. 

PROBLEM  61.  To  draw  the  monodimetric  of  a  saw-horse.  Add  the 
scale. 

These  figures  are  to  be  drawn  on  a  large  scale  so  that  each  will  fill  a 
page.  The  rear  edges  are  to  be  shown  in  dotted  lines.  The  edges  which 
divide  front  faces  from  side  faces  are  to  be  drawn  in  bold  black  lines, 
and  all  other  visible  edges  are  to  be  made  in  light  black  lines. 

35.    General  Principles  of  Isometric  Projection. 

The  difference  of  monodimetric  and  isometric  projection  has  already 
been  stated  in  paragraph  31.  It  is  this: 

In  monodimetric  projection  the  object  is  posed  so  that  two  of  its  axes 
of  dimension  are  parallel,  and  the  third  axis  perpendicular,  to  V,while  in 
isometric  projection  the  object  is  represented  so  that  all  three  axes  form 
equal  angles  with  V. 

An  isometric  drawing  of  an  object  is,  therefore,  a  real  projection, 
while  a  monodimetric  drawing  is  not. 

Careful  study  of  the  following  isometric  projections  of  a  cube  and  an 
oblong  frame  will  make  clear  these  statements. 

FIG.  46  FIG.  47. 


Every  edge  of  these  solids  is  parallel  to  one  of  the  three  axes  of  di- 
mension and  all  form  equal  angles  with  V.  It  follows  that  all  are 
shortened  equally,  i.  e.,  according  to  the  same  ratio.  They  bear  the 
same  relation  to  each  other  in  the  projection  as  they  do  in  reality. 

Having  fully  mastered  this  principle,  the  student  should  now  try  to 
understand  that,  if  the  projection  of  an  object  is  enlarged  so  as  to  make 


PROJECTION  DRAWING. 


35 


every  axis  of  dimension,  and  every  line  parallel  to  an  axis  of  dimension, 
equal  to  the  real  axis  or  line,  no  scale  will  be  needed  at  all,  i.e.,  that: 

In  an  isometric  projection  every  axial  line  (line  parallel  to  an  axis  of 
dimension)  is  directly  measurable. 

This  principle,  introduced  by  Prof.  Moellinger,  a  noted  Swiss  mathe- 
matician, is  the  one  which  has  made  isometric  projection  valuable,  because 
in  almost  every  object  of  industrial  art  the  axial  lines  abound. 

With  regard  to  position,  it  may  be  said,  that  in  an  isometric  projec- 
tion 

(1.)    The  vertical  lines  remain  vertical. 

(2.)  The  horizontal  axial  lines  form  angles  of  30°,  to  right  or  left, 
with  G  L. 

(3.)  The  length  and  position  of  non-axial  lines  may  be  found  by  mak- 
ing them  face  diagonals  or  space  diagonals  in  rectangular  blocks  formed 
by  axial  lines. 

PROBLEM  62.  To  draw  the  isometric  of  a  square  box,  open  at  the 
top.  Dimensions,  4  by  3  by  2  inches;  thickness  of  boards,  i  inch.  Show 
its  construction  and  the  nail  heads. 

36.    The  Circle  in  Isometric. 

The  circle  may  be  drawn  in  isometric  by  inclosing  it  in  a  square, 
drawing  this  in  isometric,  and  inscribing  an  ellipse. 

FIG.  48.  FIG.  49. 


Thus,  let  A  B  C  D  (Fig.  48)  be  a  square  inclosing  a  circle,  to  be  drawn 
in  isometric.  The  square  will  appear  as  a  rhombus  A  B  F  E,  having 
angles  of  60°  and  120°.  Draw  the  diameters  of  the  square  and  draw  lines 
representing  these  in  the  rhombus.  Draw  the  diagonals  in  the  square 
and  rhombus.  Find  the  points  where  the  ellipse  crosses  the  rhombus 
diagonals.  The  illustration  shows  how  these  points  may  be  found.  Draw 
the  ellipse  freehand. 

Fortunately,  this  special  form  of  the  ellipse  can  be  easily  imitated 
with  the  compasses  (oval).  An  examination  of  figure  49,  representing  a 


36 


WALTERS'  ELEMENTARY  GRAPHICS. 


cube  with  circles  inscribed  within  each  visible  face,  will  explain  the 
modus  operandi  of  this  convenient  approximation. 

Solve  the  following  problems: 

PROBLEM  63.  To  draw  the  isometric  of  a  vertical  cylinder.  Axis, 
3i  inches;  diameter,  2|  inches. 

PROBLEM  64.  To  draw  the  isometric  of  a  horizontal  cylinder  having 
the  same  dimensions. 

37.    Isometric  Projection  of  Non= Isometric  Lines. 

The  following  worked  problem  illustrates  how  non-isometric  lines  or 
figures  are  drawn  in  isometric,  and  how  the  real  length  of  non-isometric 
lines  may  be  found  by  construction. 

PROBLEM  65.  (Figs.  50  and  51).  To  make  an  isometric  drawing*;of 
an  hexagonal  prism. 

Draw  a  diagram  of  the  base  of  the  prism  inclosed  by  a  rectangle, 
A  B  C  D.  Draw  this  rectangle  in  isometric  where  it  will  form  a  rhom- 
boid with  angles  of  60°  and  12(P.  Make  E'  1'  equal  to  A  1,  2'  F'  equal  to 
2  B,  E'  6'  equal  to  A  6,  etc.  Tiiis  will  give  the  isometric  of  the  regular 
hexagon.  Then  draw  the  vertical  edges  equal  to  the  real  height  of  the 
prism.  Lastly  connect  the  upper  ends  of  the  verticals  by  lines  forming 
the  upper  isometric  hexagon. 

FIG  50.  FIG.  51. 


In  this  way  all  non-isometrical  figures  are  drawn  They  are  first  in- 
scribed within  some  figure  that  can  be  drawn  in  isometric. 

Solve  the  following  problems: 

PROBLEM  67.  To  find  the  isometric  of  a  pentagonal  prism.  Hori- 
zontal edge,  H  inch;  vertical  edge,  3  inches. 

PROBLEM  68.  To  draw  the  isometric  of  an  irregular  heptagonal 
pyramid,  from  given  orthographic  projections. 

PROBLEM  69.    To  draw  the  isometric  of  a  carpenter's  trestle. 

PROBLEM  70.  To  draw  the  isometric  of  a  tool  bench  with  braces  for 
stiffening  its  legs. 


PROJECTION   DRAWING. 


37 


38.    Cone  and  Sphere  in  Isometric. 

PROBLEM  71.    (Fig.  52).    To  draw  a  cone  in  isometric.    Diameter, 
3|  inches;  axis,  4  inches 

PROBLEM  72.    (Fig.  53)     To  make  an  isometric  drawing  of  a  hemi- 
sphere having  a  radius  of  2  inches. 

Find  the  isometric   of  the  great  circle,  as  in  the  former  problem. 
Then  draw  a  semicircle  over  the  major  axis  of  the  ellipse.    Remember 
that  a  sphere  projects  as  a  circle,  no  matter  how  it  may  be  posed 
FIG.  52.  FIG.  53. 


PROBLEM  73     To  find  the  real  radius  of  a  given  isometric  sphere. 

This  problem  is  more  difficult  lhan  might  appear  at  first.  Remem- 
berthat  only  axial  lines  ca-i  be  measured  Such  a  lin"  might  be  drawn 
through  the  center,  but  it  would  evidently  not  terminate  in  the  outline 
(principal  meridian)  of  the  sphere  Any  diameter  in  the  circle  repre- 
senting the  sphere  would  be  too  long.  In  isometric  all  figures  are  drawn 
larger  than  in  orthographic  in  order  that  their  axial  lines  may  appear  as 
they  are.  How  will  you  go  to  wqrk  to  find  it  ? 

Let  this  problem  be  a  test  for  you. 


DIVISION  H. 


SHAIDOWS. 


39.    The  Light  Parallel  to  V. 

The  form  of  the  shadow  of  an  object  depends  on: 
(1.)    The  object  casting  the  shadow,  i.  e.,  the  outline  which  the  ob- 
ject presents  to  the  light. 


38 


WALTERS'  EDEMENTARY  GRAPHICS. 


(2.)    The  surface  upon  which  the  shadow  falls. 

(3.)    The  source  or  direction  of  the  light. 

In  mechanical  drawing  the  rays  of  light  are  supposed  to  have 
some  fixed  direction,  and  in  all  examples  and  problems  of  this  division  the 
rays  of  light  are  assumed  to  be  parallel,  so  that  the  direction  of  one  ray 
is  the  direction  of  all. 

The  simplest  conditions  prevail  when  the  light  is  parallel  to  V  and 
falls  upon  H  at  an  angle  of  45°.  The  horizontal  projections  of  the  rays 
can  then  be  drawn  with  the  T-square  and  the  vertical  projections  with 
the  right-angled  isosceles  triangle.  All  shadows  will  be  on  the  right  side 
of  the  object  and  appear  in  H,  the  shadow  of  a  point  being  where  a  ray 
of  light  through  the  point  would  pierce  the  surface  upon  which  the 
shadow  is  cast. 

The  solution  of  problems  under  these  stated  conditions  is  so  simple 
that  a  study  of  figures  54  and  55  should  be  sufficient  for  the  successful 
working  by  the  student  of  all  such  problems. 

FIG.  54  FIG.  55. 


c    b 


a 


Figure  54  solves  the  problem,  to  find  the  shadow  of  a  rectangular 
block;  and  figure  55,  to  find  the  shadow  of  an  irregular  pentagonal  pyra- 
mid. 

Solve  the  following  problems  for  which  no  figures  or  solutions  are 
given: 

PROBLEM  74.  To  find  the  shadow  of  an  irregular,  obliquely  trun- 
cated pentagonal  prism. 

PROBLEM  75.    To  find  the  shadow  of  a  right  cylinder. 

This  problem  can  be  solved  by  finding  the  shadow  of  a  series  of 
points  in  the  upper  circle  and  by  connecting  these  by  a  freehand  stroke 
with  the  shadow  of  the  elments  tangent  to  the  light.  A  better  way, 
however,  is  to  find  the  shadow  of  the  upper  base,  remembering  that  the 
shadow  of  any  plane  form  cast  upon  a  parallel  plane  must  be  equal  to  the 
form,  etc. 


PROJECTION    DRAWING/. 


39 


PROBLEM  76.  To  find  the  shadow  of  a  cylinder  resting  upon  H  and 
having  its  axis  perpendicular  to  V. 

PROBLEM  77.    To  find  the  shadow  of  a  truncated  oblique  cone. 

40.    The  Light  Oblique  to  H  and  V, 

The  next  step  is  to  assume  the  rays  ot  light  oblique  to  both  planes  of 
projection.  The  simplest  conditions  of  this  kind  prevail  when  the 
descending  rays  are  assumed  so  as  to  give  horizontal  and  vertical  projec- 
tions of  45°  to  G  L.  The  student  will  observe  that  in  this  case  the  real 
angle  of  inclination  and  declination  is  not  45°,  but  that  it  is  less.  It  is 
equal  to  the  angle  which  the  space  diagonal  of  a  cube  form  with  the  base 
of  the  cube,  i.  e.,  an  angle  of  about  37|V 

The  shadow  is  cast  to  the  right  and  rear  of  the  object  and  if  this 
should  stand  close  to  V,  a  part  of  the  shadow  will  fall  upon  V.  Figure 
56  represents  a  triangle  in  space  placed  obliquely  to  V  and  H.  The 
shadow  falls  upon  both  planes  of  projection  and  forms  a  dihedral,  which 
gives  a  pentagonal  development.  The  direction  which  the  shadow  of  AB 
and  BC  takes  in  crossing  G  L  is  found  by  obtaining  th^  shadow  of  inter- 
mediate points.  Figure  57  shows  how  to  find  the  shadow  of  a  square 
prism  placed  so  that  none  of  its  lateral  faces  is  parallel  to  V,  and  that 
the  shadow  will  fall  upon  H  and  V. 

FIG.  56.  FIG.  57. 


The  student  who  is  able  to  read  every  line  of  these  two  figures 
should  have  no  great  difficulty  in  solving  the  following  problems.  Let 
him  remember  the  following: 

(1.)  The  shadow  of  a  vertical  line  forms  an  angle  of  45°  with  G  L, 
upon  H,  and  is  perpendicular  to  G  L,  upon  V. 

(2  )  The  shadow  of  a  line  perpendicular  to  V  forms  an  angle  of  45° 
with  G  L,  upon  V,  and  is  perpendicular  to  G  L,  upon  H. 


40 


WALTERS  ELEMENTARY  GRAPHICS. 


(3.)  The  shadow  of  a  line  parallel  to  V  is  equal  and  parallel  to  the 
line,  upon  V. 

(4.)  The  shadow  of  a  line  parallel  to  H  is  equal  and  parallel  to  the 
line,  upon  H. 

Solve  the  following  problems: 

PROBLEM  78.    To  find  the  shadow  cast  upon  II  of  a  cube. 

PROBLEM  79.  To  find  the  shadow  cast  upon  H  and  V  of  a  tall,  ir- 
regular pentagonal  prism. 

PROBLEM  80.  To  find  the  shadow  cast  upon  H  and  V  of  a  hexag- 
onal pyramid  standing  upon  its  apex. 

PROBLEM  81.  To  find  the  shadow  of  an  irregular  pentagonal 
pyramid  having  its  axis  parallel  to  V  and  II. 

41.        Shadows  of  Circles  and  Curved  Surfaces. 

In  problems  75,  76  and  77  the  student  has  already  solved  problems  per- 
to  taining'the  shadows  of  objects  having  curved  surfaces.  In  the  following 
problems  the  direction  of  the  light  rays  is  the  same  as  in  paragraph  40. 

FIG.  58. 


PROBLEM  82.  To  find  the  shadow  upon  H  of  a  vertical  circle  whose 
plane  is  parallel  to  V. 

PROBLEM  83.  To  find  the  shadow  upon  H  and  V  of  a  horizontal 
circle. 

PROBLEE  84.  To  find  the  shadows  in  H  and  V  of  a  circle  whose 
plane  is  perpendicular  to  G  L. 

PROBLEM  85.    To  find  the  shadow  in  H  and  V  of  a  vertical  cone. 

The  method  of  solution  (Fig.  58)  is  somewhat  peculiar.  Find  the 
shadow  of  the  apex  upon  V,  then  imagine  V  as  transparent  and  find 


PROJECTION  DRAWING. 


41 


where  the  shadow  of  the  apex  would  fall  if  such  were  the  case.  It  would 
evidently  be  cast  upon  the  extended  or  rear  part  of  H,  directly  behind  B, 
i.  e.,  at  C.  Draw  tangents  from  C  to  the  base  circle,  and  you  will  ob- 
tain the  shadow  of  the  cone  upon  H  and  its  rear  extension.  Consider 
now  V  as  opaque  and  find  the  shadow  upon  it. 

PROBLEM  86.    To  find  the  shadow  of  a  cone  standing  upon  its  apex 
near  V. 

42.    The  Shadow  of  the  Sphere. 

PROBLEM  87.    (Fig.  59.)    To  find  the  shadow  in  H  of  a  sphere. 
FIG.  59. 

I 


The  shadow  of  a  sphere  is  equal  to  the  shadow  of  the  one  of  its  great 
circles  which  is  perpendicular  to  the  rays  of  light.  If  this  great  circle  is 
found,  the  remaining  part  of  the  problem  can  present  but  few  difficulties. 
The  work  may  be  done  as  follows: 

(1.)    Find  the  shadow  of  the  center  of  the  sphere. 

(2  )  Locate  an  auxiliary  plane  perpendicular  to  H  and  parallel  to  the 
descending  rays  of  light. 

(3.)  Upon  this  plane  draw  an  auxiliary  vertical  projection  of  the 
sphere. 

(4.)  Draw  QR,  the  great  circle,  at  right  angles  to  the  rays  of  light. 
This  is  done  by  finding  the  obstructed  ray  of  light  through  the  center  A 
from  its  horizontal  projection,  and  by  drawing  Q  R  perpendicular  to  it. 

(5  )    Draw  tangent  rays  at  Q  R,  parallel  to  a    a'.    These  will  give 


42  WALTERS'  ELEMENTARY  GRAPHICS. 


the  major  diameter  of  the  ellipse  which  will  form  the  shadow.  The 
minor  diameter  will  be  equal  to  the  diameter  of  the  sphere. 

(6.)  Draw  the  ellipse  by  one  of  the  methods  given  in  paragraph  18, 
or  find  additional  points  by  the  method  shown  in  figure  59,  i.  e.,  by  using 
planes  parallel  to  H,  like  O  P. 

PROBLEM  88.    To  find  the  shadow  of  a  sphere,  cast  upon  V. 

In  monodimetric  and  isometric  projection  shades  are  often,  but 
shadows  rarely,  required.  The  rays  of  light  are  usually  assumed  so  that 
the  shadow  of  every  perpendicular  line  becomes  a  horizontal  line  equal 
in  length  to  the  perpendicular,  i.  e  ,  the  rays  are  assumed  to  come  from 
the  left,  the  front  and  above.  In  this  case  every  obstructed  ray  can  be 
drawn  with  the  right  angled  isosceles  triangle  placed  against  the  T-square. 
The  shadows  that  are  not  cast  upon  parts  of  this  object  are  assumed  to 
fall  upon  an  imaginary  horizontal  plane  supporting  the  object. 

All  shadows  given  in  this  book  are  assumed  to  be  produced  by 
straight  or  circular  outlines  and  to  be  cast  upon  H  or  V.  Book  III,  De- 
scriptive Geometry,  continues  the  subject  and  shows  how  to  find  shadows 
cast  upon  oblique  and  single  curved  surfaces  by  objects  having  more 
complex  outlines. 


PART  II. 
Shading  and  Draughting. 


43.  The  Object. 

The  object  of  this  course  in  shading  and  draughting  is  to  teach  the 
student  the  methods  of  representing  complex  objects;  to  familiarize  him 
with  the  use  of  the  scale  and  the  common  measuring  tools,  and  to  de- 
velop his  taste  with  regard  to  the  disposition  of  details,  inscriptions,  etc. 
It  is  intended  that  this  work  should  be  carried  on  simultaneously  with 
the  work  of  Part  I,  i.  e.,  that  every  alternate  lesson  should  be  given  to 
draughting  in  the  class-room  under  the  supervision  of  the  .teacher.  While 
all  work  in  graphics  must  be  done  carefully  and  neatly,  this  work,  re- 
quiring larger  sheets  of  paper  and  involving  more  complex  subjects,  will 
be  especially  suited  for  the  training  of  the  aesthetic  qualities  of  the  stu- 
dent—an educational  end  never  to  be  neglected,  though  difficult  of 
attainment. 

44.  Tools  and  Materials. 

In  addition  to  the  tools  required  for  the  home  work  and  described  in 
Part  I,  each  student  must  be  provided  with  the  following: 

FIG.  60 


(1.)  A  drawing-board.  The  board  needed  in  this  work  should  meas- 
ure 14  to  18  inches  by  ±>  to  24  inches,  and  should  be  made  of  good  quality 
of  well-seasoned  white  pine  lumber,  glued  together  of  about  four  strips. 
One  or  both  of  the  short  ends  should  be  provided  with  a  crosspiece 
(header)  of  soft  mapie,  cherry  or  some  other  small-grained  wood.  The 
header  must,  of  course,  be  straight.  Fig.  1  represents  such  a  board  with 
two  headers,  and  the  above  illustration  shows  how  a  larger  and 
better,  though  more  expensive,  board  may  be  made. 


44 


WALTERS     ELEMENTARY   GRAPHICS. 


(2.)    A  T-square ,  with  blade  extending  across  the  board.    The  simple 
construction  shown  in  Fig.  62  makes  this  form  preferable  to  any  other 

for  the  beginner. 

FIG.  61. 


(3.)  Two  triartgles,  60°,  30°,  90°,  and  45°,  90°,  respectively.  The  sides 
of  these  should  measure  not  less  than  6  inches.  Good  work  may  be  done 
with  the  right-angled  isosceles  triangle  alone.  Many  practical  draughts- 
men^use  the  tetrangle  shown  in  Fig.  63,  in  preference  to  two  triangles. 

FIG.  62. 


(4.)    Four  white  metal  thumb  tacks  to  fasten  the  sheet  to  the  drawing- 
board. 

FIG.  63. 


(5.)  Apaper  measure,  i.  e.,  a  strip  of  tough,  smooth  paper,  li  by  16 
inches,  printed  with  lines  showing  inches  and  halves,  quarters,  eighths 
and  sixteenths  of  inches.  Paper  measures  are  better  than  any  other 
cheap  measuring  tools,  and  cost  but  few  cents  per  piece. 

45.    General  Remarks. 

In  school  where  a  considerable  number  of  drawing-boards  are  used, 
it  will  be  cheaper  to  have  them  made  to  order  in  some  carpenter  shop. 
The  other  tools,  however,  should  be  bought  of  the  instrument  dealer. 
The  cheaper  grade  of  T-squares  and  triangles  are  usually  made  of  beech 
or  pear  wood;  the  best  are  the  transparent  celluloid  tools  called  amber. 


PROJECTION   DRAWING.  45 


For  storing  the  drawing-boards  between  lessons,  the  class-room  or 
an  adjoining  closet- room  ought  to  be  provided  with  a,  drawing-board  rack. 
i.  e.,  a  strong  square  case,  open  on  one  side,  into  which  the  boards  may 
be  slit  separately  between  cleats.  The  boards  should  lie  horizontally 
with  the  drawing  sheet  on  the  under  side;  the  upper  side  will  then  serve 
as  a  shelf  for  the  deposition  of  the  paper  measure  and  the  triangle  The 
top  of  the  case  should  be  of  strong  pine  lumber,  so  that  it  may  be  used  as 
a  table  for  paper  cutting  and  the  deposition  of  ink  saucers,  models,  etc. 

Wherever  the  conditions  make  it  possible,  it  is  well  to  provide  the 
drawing  room  with  top-light,  and  to  furnish  it  with  drawing  tables  that 
can  be  used  either  sitting  or  standing.  There  are  a  number  of  such 
tables  in  the  market.  The  best,  though  not  the  cheapest,  school  draw- 
ing table  known  to  the  author  of  this  book  is  the  Eugene  Dietzgen 
patent  drawing  table  I.  X.  L. 

46.    Use  of  Instruments  and  Materials. 

The  T-square  is  used  to  draw  parallel  lines  by  sliding  its  head  along 
the  left  side  of  the  drawing-board  and  using  the  upper  edge  of  the  blade 
as  a  straight  edge.  It  should  never  be  used  on  another  side  as  the  edges 
of  the  board  are  seldom  exactly  parallel  or  at  right- angles  with  each 
other.  Before  ruling  the  line,  the  blade  should  be  pressed  firmly  on  the 
paper,  as  the  square  is  liable  to  move  slight'y.  unless  the  head  of  the 
square  and  the  board  exactly  coincide.  The  square  should  be  moved 
along  the  side  of  the  board  by  the  left  hand,  leaving  the  right  hand  free 
for  the  use  of  the  pencil  or  the  pen. 

Triangles  are  used  as  straight  edges  for  drawing  perpendicular  and 
oblique  lines  to  those  already  drawn  by  the  T-square.  They  are  used 
also  for  drawing  parallel  lines,  by  placing  the  edge  of  one  along  the  line 
to  which  the  parallel  is  to  be  drawn,  and  by  sliding  the  other  triangle. 

Ink. — In  practical  draughting  India  ink  is  used,  instead  of  writing  ink. 
The  India-ink  is  preferable  in  that  it  can  stand  moisture  without 
spreading.  It  is  also  blacker  and  covers  the  paper  better,  though  it  is 
more  expensive,  especially  when  bought  jn  liquid  form.  Many  designers 
buy  India-ink  sticks  and  prepare  eac'i  dav  as  much  as  they  expect  to  use. 
Fig.  64  represents  a  stick  of  dry  India-ink. 

FIG.  64. 


To  prepare  the  India- Ink.— Rub  a  stick  of  the  ink  on  a  saucer  or  ink- 
slab  containing  about  a  teaspoonful  of  water  until  the  proper  consistency 
is  obtained.  To  determine  the  latter,  try  thejnk  from  time  to  time 
with  a  drawing-pen  on  a  piece  of  paper,  and  if,  when  di~y,  a  perfectly  black 


46  WALTERS'  ELMENTARY  GRAPHICS. 


(not  gray)  line  is  left,  the  ink  is  ready  for  use.  When  ground  it  should 
always  be  kept  covered  over,  to  prevent  its  drying  quickly,  and  so  becom- 
ing thick.  The  end  of  the  stick  of  ink  should  be  wiped  after  rubbing,  as 
otherwise  it  is  liable  to  crumble  It  is  also  a  good  plan,  when  a  long 
stick  is  used,  to  wrap  tin-foil  tightly  around  it,  to  prevent  its  thus 
breaking  or  cracking. 

Mounting  the  Paper  — Lay  the  pap^r  upon  the  drawing-board  smooth 
side  up.  Insert  one  of  the  thumb  tacks  close  to  the  edge  in  a  corner  of 
the  sheet.  Square  the  sheet  with  the  board  by  means  of  the  T-square. 
Insert  the  other  thumb  tacks. 

Ihe  Pencil  Sketch  — All  drawings  should  be  made  in  pencil  before 
inking.  The  penciled  lines  should  be  made  fine  and.  light  with  a  hard 
pencil,  so  as  to  be  easily  erased  or  inked  over.  To  erase  strong  pencil 
marks  requires  hard  rubbing,  which  destroys  the  surface  of  the  paper. 
All  penciling  should  be  done  carefully  to  avoid  confusion  in  inking. 

The  tint  lines  and  shade  lines  are  usually  drawn  in  ink  without  any 
sketching. 

Inking.— Fill  the  drawin-g  pen  by  means  of  a  writing  pen  and  keep 
the  outside  of  the  blades  clean  Hold  it  ne  irly  upright  and  carry  it  along 
the  blade  of  the  T-square  from  left  to  r  ght,  with  the  flat  sides  parallel  to 
the  direction  of  the  line.  Use  only  the  upper  edge  of  the  T-square. 
Carefully  clean  the  pen  when  through  using  In  inking  the  horizontal 
lines  should  be  first  drawn,  commencing  at  the  top,  and  then  the  vertical 
lines  commencing  at  the  left.  This  will  reduce  the  liability  to  blot  the 
work  before  it  is  dry.  Never  use  the  blotting  paper. 

47.    Inscriptions. 

It  is  important  that  the  student  of  projection  drawing  should  learn 
to  write  and  draw  neat  inscriptions.  Well-made  inscriptions  greatly 
enhance  the  beauty  of  a  drawing.  They  serve  as  an  indication  of  equal 
care  taken  in  the  execution  of  the  construction.  Tor  this  reason  the 
plate  of  the  so-called  Egyptian  capitals  and  numbers,  printed  in  Vol  I,  is 
here  reprinted,  not  to  be  copied,  but  for  reference.  There  is  also  added 
a  complete  alphabet  of  capitals,  small  letters,  and  numbers  of  round  wrik 
ing,  a  style  of  letters  so-called  because  of  its  predominant  round  form. 
On  account  of  its  distinctness,  beauty,  and  ease  of  construction,  no 
other  style  can  surpass  it  for  inscription  purposes. 

The  letters  and  figures  of  this  style  should  not  be  drawn,  but  written. 
After  some  practice  the  writing -may  be  done  with  almost  any  large  and 
elastic  pen.  It  is  best,  however,  to  procure  a  pen  made  specially  for  the 
purpose,  such  as  F.  Soennecker's  No.  2  or  3.  Larger  letters  are  made 
with  a  good  shading  pen,  or  F.  Soennecker's  Parcel's  pen  No.  133.  Stu- 
dents who  wish  to  practice  round  writing  would  do  well  to  procure  F. 
Soennecker's  Text  Book  of  Round  Writing,  published  by  the  Keuffel 
Esser  Co.,  New  York,  127  Fulton  Street. 


H 

m 

rn 
o 


PROJECTION  DRAWING.  49 

For  many  purposes  a  written  plain  vertical  letter  will  answer  best. 
48.    Blue-printing  and  Black-printing. 

The  invention  of  the  beautiful  processes  of  blue  or  black-printing  and 
photo  zinc  etching  has,  of  late,  greatly  changed  the  methods  of  work  in 
practical  draughting.  Formerly  most  drawings  were  finished  in  colors 
applied  with  the  brush,  but  at  present  nearly  all  ordinary  work  is  being 
done  in  black  lines. 

In  order  to  reproduce  a  drawing  by  the  blue  or  black-printing  pro- 
cess, it  is  first  made  in  pencil  the  same  as  every  other  drawing.  It  is 
then  traced  in  strong  India-ink  upon  a  sheet  of  transparent  paper  called 
tracing  paper  or  patera. 

The  next  step  is  to  coat  the  blue-print  paper  with  sensitive  material. 
The  paper  is  simply  common  drawing  or  writing  paper,  usually  French 
satin.  The  material  is  a  mixture  in  nearly  equal  parts  of  dissolved 
cit.  iron  and  ammonium  and  pot.  perrocyanide.  These  chemicals  are  dis- 
solved separately  in  clean  water,  in  a  ratio  of  about  two  tablespoons  of 
each  compound,  finely  ground,  to  one  cup  of  water,  and  allowed  to  stand 
in  bottles  in  a  dark  place  for  about  forty-eight  hours.  The  paper  is  then 
tacked  to  a  wall  in  an  entirely  dark  room,  the  two  solutions  are  poured 
into  a  cup,  and  the  mixture  is  carefully  applied  to  the  paper  with  a  large, 
smooth  paint-brush  or  a  sponge.  It  dries  rapidly  and  will  be  ready  for 
use  in  two  to  three  hours. 

The  printing  is  done  as  follows:  The  sheet  of  sensitized  paper  is 
laid,  coated  side  up,  upon  a  drawing-board;  the  tracing  is  laid  over  it; 
a  plate  of  thick  glass  is  laid  over  the  latter  to  hold  it  down  at  every  point, 
and  the  whole  is  exposed  to  the  bright  sunlight  for  about  five  minutes, 
when  the  whole  surface  of  the  coated  sheet  will  have  turned  from  a  light 
yellow  to  a  dirty  bluish  gray.  It  is  then  immediately  immersed  in  a  bath 
of  clean  water.  The  water  will  wash  off  the  coating  where  it  was  not 
fixed  by  the  sunlight,  i.  e  ,  under  the  black  lines  of  the  tracing,  and  these 
will  appear  white,  while  the  rest  of  the  surface  will  turn  a  brilliant  blue. 
It  is  evident  that  as  many  copies  can  be  made  of  a  tracing  as  may  be 
desired.  In  architects'  and  engineers'  draughting  rooms  more  complex 
apparatus  is  used  than  the  one  here  described,  but  the  process  is  the 
same. 

Formerly  every  artisan  or  artist  prepared  his  own  paper.  For  the 
last  half-dozen  years,  however,  factories  of  sensitized  paper  have  been 
established  in  all  large  cities,  and  carefully  prepared  "  blue  paper"  costs 
now  but  little  more  than  other  paper.  The  difficulty  with  "  ready  blue 
paper  "  is  in  that  it  must  be  kept  in  a  dark  place.  An  exposure  to  sun- 
light of  only  a  few  seconds  will  start  the  chemical  action  and  render  it 
valueless.  All  schools  of  applied  mathematics  have  blue-printing  rooms 
connected  with  their  classrooms  in  draughting,  and  finish  their  work  in 
white  upon  blue  ground. 

The  black-printing  process  does  not  differ  from   the  blue-printing 


50  WALTERS'  ELEMENTARY  GRAPHICS. 


process  in  the  manipulation  of  the  drawing.  The  lines  appear  black 
upon  light  gray  or  mottled  ground.  The  paper  is  difficult  to  prepare  and 
is,  therefore,  bought  ready  for  printing.  The  materials  cost  more  than 
those  used  in  blue-printing.  As  a  result  the  process  is  used  almost  only 
by  sculupturers  and  woodcarvers  in  whose  drawings  the  blue-printing 
process  would  reverse  light  and  shade. 

Photo-zinc  etching  requires  a  considerable  amount  of  costly  apparatus, 
and  the  work  of  etching,  though  it  is  a  purely  chemical  manipulation, 
requires  a  good  deal  of  skill.  It  is,  therefore,  done  only  in  special  art 
establishments.  The  product,  a  relief  plate,  can  be  used  in  every  print- 
ing press.  Nearly  all  line  pictures  in  the  books  and  newspapers  of  to-day 
are  printed  from  photo-zinc  etchings  made  from  pen  drawings,  and  every 
well-made  pen  drawing  may  thus  be  photograved.  All  the  illustrations  of 
this  book  were  printed  from  photo-zinc  etching  made  from  pen  drawings 
most  of  which  were  prepared  by  L.  P.  Brous.  student  assistant  in  the 
Department  of  Industrial  Art  at  the  Kansas  State  Agricultural  College. 

49.    Shades. 

PLATE  I.  This  figure  represents  the  vertical  and  horizontal  projec- 
tions of  a  group  of  solids  consisting  of  a  square  slab,  broken  off  behind,  an 
octagonal  prism,  a  cylinder  and  an  octagonal  pyramid. 

The  group  is  placed  so  that  the  source  of  light  is  to  the  front,  the  left 
and  above,  and  the  shadow  to  the  rear  and  right.  Some  of  the  faces  of 
the  group,  especially  of  the  pyramid,  are  nearly  at  right  angles  to  the 
light  rays,  and  receive  evidently  more  light,  proportionately,  than  the 
horizontal  surfaces,  or  some  of  the  faces  of  the  prism  and  parts  of  the 
cylinder.  Along  two  of  the  elements  of  the  cylinder  the  light  rays  are 
tangent  only,  i.  e  ,  they  merely  graze  the  object  The  parts  of  the  solids 
on  the  opposite  or  rear  side  of  the  group  receive  only  reflected,  but  no 
direct,  light.  It  is  these  different  light  effects,  which  may  be  watched 
on  every  object,  that  the  shaded  drawing  represents. 

In  order  to  shade  an  object  properly,  the  student  must  be  able  to 
estimate  approximately  the  quantity  of  light  received  by  every  part  of 
the  surface,  or,  which  is  the  same,  the  angle  which  every  part  of  the  sur- 
face forms  with  the  light  rays. 

There  are,  however,  a  few  exceptions  or  modifications  to  this  rule. 
In  the  first  place,  there  is  the  influence  of  the  distance  of  the  object  from 
the  eye.  Things  near  the  eye  display  more  contrasts  of  light  and  shade 
than  things  farther  away,  while  at  any  great  distance  all  shades  and 
lights,  and  even  colors,  seem  to  blend  into  one  general  tone.  This  is  due 
to  the  fact  that  object^  appear  smaller  at  a  distance,  and  that  the  atmos- 
phere is  slightly  opaque.  It  may  be  said  then,  that  dark  surfaces  grow 
lighter  and  light  surfaces  darker  as  they  recede.  In  projection  drawing  it  is 
necessary  to  greatly  exaggerate  the  difference  of  shade  between  the  nearer 
and  farther  parts  of  a  surface  in  order  to  show  the  difference  of  distance 
from  the  eye. 


PROJECTION   DRAWING. 


51 


52  WALTERS'   ELEMENTARY   GRAPHICS. 


In  the  second  place,  there  is  the  influence  of  contrast.  The  light  parts 
appear  lighter  near  dark  parts,  and  the  dark  parts  seem  darker  close  to 
light  parts.  This  fact  is  well-known  by  every  one  and  is  taken  advantage 
of  in  decorating  and  drawing.  As  a  consequence  the  shading  of  the 
group  of  solids  is  strongest  where  the  shaded  surfaces  are  adjacent  to 
light  surfaces,  and  vice  versa. 

Upon  objects  having  curved  surfaces  like  the  cylinder,  the  depth  of 
shade  must  change  at  every  element.  The  brightest  part  is  a  little 
forward  of  the  point  in  H  where  the  horizontal  projection  of  a  ray  of 
light  through  the  axis  of  the  cylinder  would  pierce  the  circle  which  forms 
the  horizontal  projection  of  the  cylinder.  This  part  of  the  curved  sur- 
face is  called  the  brilliant  line.  It  is  placed  forward  of  the  element  which 
reclines  the  most  to  the  light,  because  the  light  of  this  element,  ac- 
cording to  a  well-known  law  in  physics,  would  be  reflected  back  from 
whence  it  came,  while  the  light  from  elements  a  little  more  to  the  right 
would  be  reflected  forward  in  the  direction  of  the  observer.  The  latter 
elements,  though  receiving  less  light,  would  really  appear  lighter. 

The  darkest  parts  of  the  surface  are  the  elements  nearest  the  point 
of  tangency.  The  surface  from  to  the  outline  at  the  right  should  appar- 
ently be  of  the  same  shade,  being  all  in  the  shadow.  But  here  another 
principle  has  to  be  observed.  There  is  always  a  small  amount  of  light 
coming  in  exactly  the  opposite  direction  from  the  orignal  one.  This  is 
called  reflected  light,  and,  owing  to  this,  the  surface  to  the  right  is  made 
a  little  lighter  as  it  turns  back.  See  figure  68. 

This  principle  influences  also  the  shading  of  the  pyramid. 

A  careful  study  of  the  given  group  of  solids  will  reveal  a  number  of 
applications  of  the  exceptions  mentioned.  The  student  should  not  draw 
a  single  line,  the  meaning  of  which  he  does  not  understand.  In  case  he 
cannot  find  reasons  for  drawing  certain  lines,  he  should  ask  the  eacher 
for  an  explanation. 

The  outlines  of  the  shadows  are  found  by  drawing  at  45°  to  G  L, 
tangent  to  the  outlines  in  H.  The  shadow  of  the  pyramid  is  found  by 
establishing  the  shadow  of  the  apex  uuon  the  level  of  the  base  of  the 
pyramid,  and  drawing  from  there,  in  H,  tangent  to  the  base  of  this  solid. 

The  figure  should  be  copied  twice  its  linear  dimensions,  i.  e.,  four 
times  as  large  as  the  given  original.  The  outline  should  be  carefully  laid 
out  in  pencil  and  inked.  The  shade  lines  should  be  drawn  without  any 
sketching  and  should  be  made  as  near  as  possible  like  the  original  in 
thickness— not  stronger.  The  figure  being  larger,  there  should  be  an  in- 
creased number  of  shade  lines. 

All  outlines  separating  light  from  dark  surfaces  should  be  drawn 
considerably  stronger  than  those  separating  faces  of  equal  or  nearly  equal 
light  effects.  This  rule  should  be  followed  in  all  mechanical  drawing, 
and  must  not  be  neglected  in  drawing  the  problems  of  Part  I,  though 
nothing  is  said  there  about  shading. 

Provide  the  plate  with  a  uniform  border  line. 


PROJECTION   DRAWING. 


53 


In  the  upper  left  corner  of  the  plate  write  a  neat  inscription  in  round 
writing,  as  described  in  paragraph  47. 

In  the  lower  right  corner  add  your  signature. 


FIG.  68. 
50.    Shades  upon  Complex  Surfaces. 

PLATE  II.  This  plate  represents  a  group  of  solids  consisting  of  tfcN^& 
well-known  geometric  forms  and  a  part  of  the  shaft  of  a  fluted  column. 

The  direction  of  the  light  is  the  same  as  in  the  last  plate,  and  there 
are  no  new  principles  involved.  The  solids,  however,  are  more  complex 
and  require  a  more  subtle  grading  of  shade  lines.  The  strongest  shade 
lines  of  the  fluted  shaft  and  the  cone  are  made  by  drawing  two  light 


54 


WALTERS    ELEMENTARY  GRAPHICS. 


PROJECTION   DRAWING.  55 


outlines  for  each  and  filling  the  space  between  the  two  by  drawing  an 
other  line  or  two  with  the  drawing  pen.  To  shade  such  a  plate  well 
requires  careful  study  of  the  principles  involved  and  a  close  application 
in  the  work. 

Draw  the  figure  not  less  than  two  diameters  of  the  original. 

Add  a  neat  border  line. 

ILI  the  upper  left  corner  inside  the  border  line  write  a  neat  inscription 
in  round  writing,  as  described  in  paragraph  47. 

In  the  lower  right  corner  sign  your  name. 

51.    Shades  upon  Double  Curved  Surfaces. 

PLATE  III  shows  a  cavetto,  a  sphere  and  a  cone,  shaded  by  means 
of  irregularly  placed  points  or  dots.  Such  shading  or  tinting  is  called 
stippling.  It  is  done  with  the  common  writing  pen,  and  is  not  difficult  to 
learn.  The  principles  of  shading  are,  of  course,  the  same,  no  matter 
what  graphic  method  may  be  followed  in  producing  the  desired  effect. 

The  student  should  take  care  not  to  place  the  dots  in  rows,  but  to 
distribute  them  evenly  over  the  surface,  much  as  the  farmer  sows  the 
seed  upon  a  field,  i.  e  ,  irregular  in  position,  but  regular  in  distribution. 

Note  that  the  sphere  has  its  brilliant  point  not  where  the  light  is 
perpendicular  to  the  surface,  but  somewhat  nearer  the  center  of  the  pro- 
jection. The  reasons  for  this  are  those  given  with  regard  to  the  shading 
of  the  cylinder  in  plate  1.  The  lisrht  rays  are  tangent  to  the  sphere  along 
a  great  circle  which  projects  in  H  and  V  as  an  ellipse.  The  rear  surface 
of  the  sphere  is  somewhat  lighter  than  the  region  where  the  light  is 
tangent,  for  reasons  already  stated. 

The  shadows  of  the  cone  upon  the  sphere  aiid  the  shadow  of  the 
sphere  upon  the  scotia  is  not  expected  to  be  found  by  the  pupil,  but 
should  be  copied  from  the  original  plate.  The  proper  solution  of  these 
two  problems  must  be  left  to  higher  descriptive  geometry.  Remember  that 
all  shadows  appear  strongest  along  the  outline  and  lighter  toward  the 
middle. 

Fig.  68,  which  represents  a  carpenter's  glue-pot  with  glue  brushes  in 
it,  taken  from  Walter's  Object  Drawing,  shows  how  shading  of  double 
curved  surfaces  is  being  done  in  freehand  pen  work.  This  method  o^ 
shading  is  sometimes  used  in  draughting  to  represent  objects  having 
irregular  or  broken  surfaces,  such  as  rough  cut  stone  or  carved  woodi 
while  stippling  is  used  for  smooth  surfaces  like  paper,  leather, 
unpolished  metal,  etc.  Fig.  68,  belonging  to  the  course  of  advanced 
freehand  drawing,  is  reprinted  here  for  comparison  only,  and  is  not  to  be 
copied. 

Draw  the  plate  twice  as  high  and  wide,  i.  e.,  four  times  as  large  as 
the  given  original. 

Draw  a  neat  border  line. 

Write  an  inscription  in  the  upper  left  corner,  using  the  round  writing 

letter. 

Add  your  signature  in  the  lower  left  corner. 


56 


WALTERS'  ELEMENTARY  GRAPHICS. 


PROJECTION   DRAWING. 


57 


52.    Shaded  Isometric  Projections. 

PLATE  IV.  This  plate  is  to  be  given  to  shading  of  isometric  projec- 
tions. There  will  be  two  figures,  to  be  drawn  as  follows: 

Figure  A  is  to  be  an  enlarged  copy  of  the  given  cubical  frame ;  side, 
2*  inches. 

Figure  B  is  to  represent  an  isometric  cube  having  a  side  of  H  inches. 
Upon  each  face  draw  a  square  pyramid  having  an  axis  of  H  inches  and  a 
base  of  1  inch. 

Ink  the  visible  outline  of  both  figures  and  erase  all  non-visible  parts. 

FIG.  71. 


Shade  the  edges  dividing  light  from  dark  faces,  assuming  the  light  to 
come  from  the  left  and  above. 

Shade  all  faces  to  right  by  strong  tintlines  placed  close.  All  shaded 
faces  but  two  of  the  second  figure,  will  be  vertical  and  must  be  shaded 
by  vertical  tint  lines.  The  two  faces  in  the  second  figure,  which  are  not 
vertical,  are  the  right  faces  of  the  top  and  bottom  pyramids.  These  are 
to  be  shaded  in  the  direction  of  the  altitude  of  the  face  triangles. 

Ink  the  border  line. 

Add  an  inscription. 

Sign  your  name. 

53.    Plan  of  a  Home  lot. 

PLATE  V.  This  plate  is  to  consist  of  an  enlarged  copy  of  figure  72 
(adopted,  with  some  alterations,  from  Prof.  W.  B.  Ware's  Examples  of 
Building  Construction). 

The  given  plan  is  drawn  to  a  scale  of  1  inch  to  32  feet,  but  the  copy 
is  to  be  drawn  four  times  as  large,  i.e.,  to  a  scale  of  1 : 192.  Remember 
that  plane  figures  are  to  each  other  as  the  squares  of  homologous  lines. 

Turn  the  plan  around  so  that  north  will  be  above  and  south  below, 
which  is  the  usual  position  of  a  landscape  plan.  Sketch  the  roads  and 
the  outlines  of  the  tree-clumps  in  pencil,  measuring  all  positions  by  tri- 
angulation  from  the  base  line.  Finish  the  former  with  hard-rubber 


58 


WALTERS     ELEMENTARY   GRAPHICS. 


curve  and    drawing    pen,  and    draw    the    latter    freehand,    using  the 
writing  pen.    Notice  how  the  shading  of  the  clumps  and  ground  on  the 
east  gives  relief  and  naturalness  to  the  lawn. 
Draw  a  border  line. 

FIG.  72 


PROJECTION   DRAWING. 


59 


Add  a  neat  inscription,  "Homelot"  over  the  plan.  This  may  be 
drawn  in  the  simple  and  easily  sketched  Egyptian  letter,  but  it  should 
not  be  enlarged,  and  the  letters  should  be  placed  close  together. 

In  the  left  corner,  below,  write  in  small,  but  neat  letters,  "  Scale  1 
Inch  to  16  Feet." 

In  the  right  corner  sign  your  name. 

54.    A  Floor  Plan. 

PLATE  VI.  This  plate  is  to  consist  of  an  enlarged  copy  of  the  given 
floor  plan  (taken,  with  some  alterations,  from  Soennecker's  Textbook  on 
Round  Writing.) 

The  given  plan  is  drawn  to  the  scale  of  1  inch  to  16  feet.  Make  the 
copy  four  times  as  large,  i.  e  ,  to  the  scale  of  1  inch  to  8  feet. 

FIG.  73. 


60  WALTERS'  ELEMENTARY  GRAPHICS. 


Plastered  partitions  on  studs  are  assumed  to  be  6  inches  in  thick, 
ness,  half  partitions  on  each  side  of  a  sliding  door,  3  inches;  and  the 
space  for  the  sliding  door,  3  inches.  Brick  walls  are  usually  made  12 
inches  in  thickness.  The  line  outside  the  wall  represents  the  watertable, 
a  stone  band  even  with  the  floor,  which  projects  about  2  inches.  Door 
openings  are  left  blank,  and  windows  are  shown  by  3  lines  connecting 
two  squares  that  represent  boxjambs  and  are  partly  inserted  in  the  brick 
walls. 

After  copying  this  floor  plan  it  would  be  good  practice  for  the 
student  to  draw  a  set  of  floor  plans  of  his  home,  from  measurements 
taken  by  himself.  This  would  give  him,  in  addition,  an  insight  into  the 
work  of  architectural  construction,  which  mere  copying  of  plans  and  eleva- 
tions cannot  give. 

All  inscriptions  should  be  horizontal,  straight,  plain  and  small. 
There  should  be  no  attempt  at  fancy  work  of  any  kind. 

Always  add  scale  and  signature  to  mechanical  drawings  of  this  kind. 


FOURTEEN  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 


This  book  is  due  on  the  last  date  stamped  below,  or 

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28Nov'55PTZ 

NOV291955LU 

-*ji$* 

\v" 

REC'D  LD 

IIA3  1  0  1962 

LD  21-100m-2  '55                                    T    .General  Library              / 
(B139s22)476'                                        University  of  California      / 

YC  66,27 


M319099 


